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Theorem dfrdg4 36482
Description: A quantifier-free definition of the recursive definition generator. (Contributed by Scott Fenton, 17-Apr-2014.) (Revised by Mario Carneiro, 19-Apr-2014.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
Assertion
Ref Expression
dfrdg4 rec(𝐹, 𝐴) = ∪ (( Funs ∩ (◡Domain “ On)) ∖ dom ((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))))))
Proof of Theorem dfrdg4
Dummy variables 𝑎 𝑏 𝑓 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfrdg3 36325 . 2 rec(𝐹, 𝐴) = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))}
2 an12 658 . . . . . . . 8 ((𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))) ↔ (𝑓 Fn 𝑥 ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))))
3 df-fn 6543 . . . . . . . . . 10 (𝑓 Fn 𝑥 ↔ (Fun 𝑓 ∧ dom 𝑓 = 𝑥))
4 ancom 466 . . . . . . . . . 10 ((Fun 𝑓 ∧ dom 𝑓 = 𝑥) ↔ (dom 𝑓 = 𝑥 ∧ Fun 𝑓))
5 eqcom 2772 . . . . . . . . . . 11 (dom 𝑓 = 𝑥 ↔ 𝑥 = dom 𝑓)
65anbi1i 636 . . . . . . . . . 10 ((dom 𝑓 = 𝑥 ∧ Fun 𝑓) ↔ (𝑥 = dom 𝑓 ∧ Fun 𝑓))
73, 4, 63bitri 300 . . . . . . . . 9 (𝑓 Fn 𝑥 ↔ (𝑥 = dom 𝑓 ∧ Fun 𝑓))
87anbi1i 636 . . . . . . . 8 ((𝑓 Fn 𝑥 ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))) ↔ ((𝑥 = dom 𝑓 ∧ Fun 𝑓) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))))
9 anass 474 . . . . . . . 8 (((𝑥 = dom 𝑓 ∧ Fun 𝑓) ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))) ↔ (𝑥 = dom 𝑓 ∧ (Fun 𝑓 ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))))
102, 8, 93bitri 300 . . . . . . 7 ((𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))) ↔ (𝑥 = dom 𝑓 ∧ (Fun 𝑓 ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))))
1110exbii 1881 . . . . . 6 (∃𝑥(𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))) ↔ ∃𝑥(𝑥 = dom 𝑓 ∧ (Fun 𝑓 ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))))
12 vex 3461 . . . . . . . 8 𝑓 ∈ V
1312dmex 7912 . . . . . . 7 dom 𝑓 ∈ V
14 eleq1 2853 . . . . . . . . 9 (𝑥 = dom 𝑓 → (𝑥 ∈ On ↔ dom 𝑓 ∈ On))
15 raleq 3322 . . . . . . . . 9 (𝑥 = dom 𝑓 → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) ↔ ∀𝑦 ∈ dom 𝑓(𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
1614, 15anbi12d 644 . . . . . . . 8 (𝑥 = dom 𝑓 → ((𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))) ↔ (dom 𝑓 ∈ On ∧ ∀𝑦 ∈ dom 𝑓(𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))))
1716anbi2d 642 . . . . . . 7 (𝑥 = dom 𝑓 → ((Fun 𝑓 ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))) ↔ (Fun 𝑓 ∧ (dom 𝑓 ∈ On ∧ ∀𝑦 ∈ dom 𝑓(𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))))
1813, 17ceqsexv 3505 . . . . . 6 (∃𝑥(𝑥 = dom 𝑓 ∧ (Fun 𝑓 ∧ (𝑥 ∈ On ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))) ↔ (Fun 𝑓 ∧ (dom 𝑓 ∈ On ∧ ∀𝑦 ∈ dom 𝑓(𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))))
1911, 18bitri 278 . . . . 5 (∃𝑥(𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))) ↔ (Fun 𝑓 ∧ (dom 𝑓 ∈ On ∧ ∀𝑦 ∈ dom 𝑓(𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))))
20 df-rex 3092 . . . . 5 (∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))) ↔ ∃𝑥(𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))))
21 eldif 3916 . . . . . 6 (𝑓 ∈ (( Funs ∩ (◡Domain “ On)) ∖ dom ((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))))) ↔ (𝑓 ∈ ( Funs ∩ (◡Domain “ On)) ∧ ¬ 𝑓 ∈ dom ((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))))))
22 elin 3922 . . . . . . . 8 (𝑓 ∈ ( Funs ∩ (◡Domain “ On)) ↔ (𝑓 ∈ Funs ∧ 𝑓 ∈ (◡Domain “ On)))
2312elfuns 36444 . . . . . . . . 9 (𝑓 ∈ Funs ↔ Fun 𝑓)
2412elima 6069 . . . . . . . . . 10 (𝑓 ∈ (◡Domain “ On) ↔ ∃𝑥 ∈ On 𝑥◡Domain𝑓)
25 df-rex 3092 . . . . . . . . . 10 (∃𝑥 ∈ On 𝑥◡Domain𝑓 ↔ ∃𝑥(𝑥 ∈ On ∧ 𝑥◡Domain𝑓))
26 vex 3461 . . . . . . . . . . . . . . 15 𝑥 ∈ V
2726, 12brcnv 5870 . . . . . . . . . . . . . 14 (𝑥◡Domain𝑓 ↔ 𝑓Domain𝑥)
2812, 26brdomain 36462 . . . . . . . . . . . . . 14 (𝑓Domain𝑥 ↔ 𝑥 = dom 𝑓)
2927, 28bitri 278 . . . . . . . . . . . . 13 (𝑥◡Domain𝑓 ↔ 𝑥 = dom 𝑓)
3029anbi1ci 638 . . . . . . . . . . . 12 ((𝑥 ∈ On ∧ 𝑥◡Domain𝑓) ↔ (𝑥 = dom 𝑓 ∧ 𝑥 ∈ On))
3130exbii 1881 . . . . . . . . . . 11 (∃𝑥(𝑥 ∈ On ∧ 𝑥◡Domain𝑓) ↔ ∃𝑥(𝑥 = dom 𝑓 ∧ 𝑥 ∈ On))
3213, 14ceqsexv 3505 . . . . . . . . . . 11 (∃𝑥(𝑥 = dom 𝑓 ∧ 𝑥 ∈ On) ↔ dom 𝑓 ∈ On)
3331, 32bitri 278 . . . . . . . . . 10 (∃𝑥(𝑥 ∈ On ∧ 𝑥◡Domain𝑓) ↔ dom 𝑓 ∈ On)
3424, 25, 333bitri 300 . . . . . . . . 9 (𝑓 ∈ (◡Domain “ On) ↔ dom 𝑓 ∈ On)
3523, 34anbi12i 640 . . . . . . . 8 ((𝑓 ∈ Funs ∧ 𝑓 ∈ (◡Domain “ On)) ↔ (Fun 𝑓 ∧ dom 𝑓 ∈ On))
3622, 35bitri 278 . . . . . . 7 (𝑓 ∈ ( Funs ∩ (◡Domain “ On)) ↔ (Fun 𝑓 ∧ dom 𝑓 ∈ On))
3736anbi1i 636 . . . . . 6 ((𝑓 ∈ ( Funs ∩ (◡Domain “ On)) ∧ ¬ 𝑓 ∈ dom ((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))))) ↔ ((Fun 𝑓 ∧ dom 𝑓 ∈ On) ∧ ¬ 𝑓 ∈ dom ((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))))))
38 brdif 5166 . . . . . . . . . . . . . . 15 (𝑓((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))))𝑦 ↔ (𝑓(◡ E ∘ Domain)𝑦 ∧ ¬ 𝑓 Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))))𝑦))
39 vex 3461 . . . . . . . . . . . . . . . . . 18 𝑦 ∈ V
4012, 39brco 5858 . . . . . . . . . . . . . . . . 17 (𝑓(◡ E ∘ Domain)𝑦 ↔ ∃𝑥(𝑓Domain𝑥 ∧ 𝑥◡ E 𝑦))
4128anbi1i 636 . . . . . . . . . . . . . . . . . . 19 ((𝑓Domain𝑥 ∧ 𝑥◡ E 𝑦) ↔ (𝑥 = dom 𝑓 ∧ 𝑥◡ E 𝑦))
4241exbii 1881 . . . . . . . . . . . . . . . . . 18 (∃𝑥(𝑓Domain𝑥 ∧ 𝑥◡ E 𝑦) ↔ ∃𝑥(𝑥 = dom 𝑓 ∧ 𝑥◡ E 𝑦))
43 breq1 5114 . . . . . . . . . . . . . . . . . . 19 (𝑥 = dom 𝑓 → (𝑥◡ E 𝑦 ↔ dom 𝑓◡ E 𝑦))
4413, 43ceqsexv 3505 . . . . . . . . . . . . . . . . . 18 (∃𝑥(𝑥 = dom 𝑓 ∧ 𝑥◡ E 𝑦) ↔ dom 𝑓◡ E 𝑦)
4542, 44bitri 278 . . . . . . . . . . . . . . . . 17 (∃𝑥(𝑓Domain𝑥 ∧ 𝑥◡ E 𝑦) ↔ dom 𝑓◡ E 𝑦)
4613, 39brcnv 5870 . . . . . . . . . . . . . . . . . 18 (dom 𝑓◡ E 𝑦 ↔ 𝑦 E dom 𝑓)
4713epeli 5565 . . . . . . . . . . . . . . . . . 18 (𝑦 E dom 𝑓 ↔ 𝑦 ∈ dom 𝑓)
4846, 47bitri 278 . . . . . . . . . . . . . . . . 17 (dom 𝑓◡ E 𝑦 ↔ 𝑦 ∈ dom 𝑓)
4940, 45, 483bitri 300 . . . . . . . . . . . . . . . 16 (𝑓(◡ E ∘ Domain)𝑦 ↔ 𝑦 ∈ dom 𝑓)
5049anbi1i 636 . . . . . . . . . . . . . . 15 ((𝑓(◡ E ∘ Domain)𝑦 ∧ ¬ 𝑓 Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))))𝑦) ↔ (𝑦 ∈ dom 𝑓 ∧ ¬ 𝑓 Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))))𝑦))
5138, 50bitri 278 . . . . . . . . . . . . . 14 (𝑓((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))))𝑦 ↔ (𝑦 ∈ dom 𝑓 ∧ ¬ 𝑓 Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))))𝑦))
52 onelon 6389 . . . . . . . . . . . . . . . . . . . . . . . 24 ((dom 𝑓 ∈ On ∧ 𝑦 ∈ dom 𝑓) → 𝑦 ∈ On)
53523adant1 1148 . . . . . . . . . . . . . . . . . . . . . . 23 ((Fun 𝑓 ∧ dom 𝑓 ∈ On ∧ 𝑦 ∈ dom 𝑓) → 𝑦 ∈ On)
54 brun 5164 . . . . . . . . . . . . . . . . . . . . . . . . 25 (⟨𝑓, 𝑦⟩(((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))𝑥 ↔ (⟨𝑓, 𝑦⟩((V × {∅}) × {∪ {𝐴}})𝑥 ∨ ⟨𝑓, 𝑦⟩((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))𝑥))
55 brxp 5712 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (⟨𝑓, 𝑦⟩((V × {∅}) × {∪ {𝐴}})𝑥 ↔ (⟨𝑓, 𝑦⟩ ∈ (V × {∅}) ∧ 𝑥 ∈ {∪ {𝐴}}))
56 opelxp 5699 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (⟨𝑓, 𝑦⟩ ∈ (V × {∅}) ↔ (𝑓 ∈ V ∧ 𝑦 ∈ {∅}))
5712, 56mpbiran 722 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (⟨𝑓, 𝑦⟩ ∈ (V × {∅}) ↔ 𝑦 ∈ {∅})
58 velsn 4607 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
5957, 58bitri 278 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (⟨𝑓, 𝑦⟩ ∈ (V × {∅}) ↔ 𝑦 = ∅)
60 velsn 4607 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑥 ∈ {∪ {𝐴}} ↔ 𝑥 = ∪ {𝐴})
6159, 60anbi12i 640 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((⟨𝑓, 𝑦⟩ ∈ (V × {∅}) ∧ 𝑥 ∈ {∪ {𝐴}}) ↔ (𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}))
6255, 61bitri 278 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (⟨𝑓, 𝑦⟩((V × {∅}) × {∪ {𝐴}})𝑥 ↔ (𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}))
63 brun 5164 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (⟨𝑓, 𝑦⟩((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))𝑥 ↔ (⟨𝑓, 𝑦⟩(( Bigcup ∘ Img) ↾ (V × Limits ))𝑥 ∨ ⟨𝑓, 𝑦⟩((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))𝑥))
6426brresi 5989 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (⟨𝑓, 𝑦⟩(( Bigcup ∘ Img) ↾ (V × Limits ))𝑥 ↔ (⟨𝑓, 𝑦⟩ ∈ (V × Limits ) ∧ ⟨𝑓, 𝑦⟩( Bigcup ∘ Img)𝑥))
65 opelxp 5699 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (⟨𝑓, 𝑦⟩ ∈ (V × Limits ) ↔ (𝑓 ∈ V ∧ 𝑦 ∈ Limits ))
6612, 65mpbiran 722 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (⟨𝑓, 𝑦⟩ ∈ (V × Limits ) ↔ 𝑦 ∈ Limits )
6739ellimits 36439 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 ∈ Limits ↔ Lim 𝑦)
6866, 67bitri 278 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (⟨𝑓, 𝑦⟩ ∈ (V × Limits ) ↔ Lim 𝑦)
69 opex 5447 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⟨𝑓, 𝑦⟩ ∈ V
7069, 26brco 5858 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (⟨𝑓, 𝑦⟩( Bigcup ∘ Img)𝑥 ↔ ∃𝑧(⟨𝑓, 𝑦⟩Img𝑧 ∧ 𝑧 Bigcup 𝑥))
71 vex 3461 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 𝑧 ∈ V
7212, 39, 71brimg 36466 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (⟨𝑓, 𝑦⟩Img𝑧 ↔ 𝑧 = (𝑓 “ 𝑦))
7326brbigcup 36427 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑧 Bigcup 𝑥 ↔ ∪ 𝑧 = 𝑥)
7472, 73anbi12i 640 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((⟨𝑓, 𝑦⟩Img𝑧 ∧ 𝑧 Bigcup 𝑥) ↔ (𝑧 = (𝑓 “ 𝑦) ∧ ∪ 𝑧 = 𝑥))
7574exbii 1881 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (∃𝑧(⟨𝑓, 𝑦⟩Img𝑧 ∧ 𝑧 Bigcup 𝑥) ↔ ∃𝑧(𝑧 = (𝑓 “ 𝑦) ∧ ∪ 𝑧 = 𝑥))
7612imaex 7917 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑓 “ 𝑦) ∈ V
77 unieq 4885 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑧 = (𝑓 “ 𝑦) → ∪ 𝑧 = ∪ (𝑓 “ 𝑦))
7877eqeq1d 2767 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑧 = (𝑓 “ 𝑦) → (∪ 𝑧 = 𝑥 ↔ ∪ (𝑓 “ 𝑦) = 𝑥))
7976, 78ceqsexv 3505 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (∃𝑧(𝑧 = (𝑓 “ 𝑦) ∧ ∪ 𝑧 = 𝑥) ↔ ∪ (𝑓 “ 𝑦) = 𝑥)
80 eqcom 2772 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (∪ (𝑓 “ 𝑦) = 𝑥 ↔ 𝑥 = ∪ (𝑓 “ 𝑦))
8179, 80bitri 278 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (∃𝑧(𝑧 = (𝑓 “ 𝑦) ∧ ∪ 𝑧 = 𝑥) ↔ 𝑥 = ∪ (𝑓 “ 𝑦))
8270, 75, 813bitri 300 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (⟨𝑓, 𝑦⟩( Bigcup ∘ Img)𝑥 ↔ 𝑥 = ∪ (𝑓 “ 𝑦))
8368, 82anbi12i 640 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((⟨𝑓, 𝑦⟩ ∈ (V × Limits ) ∧ ⟨𝑓, 𝑦⟩( Bigcup ∘ Img)𝑥) ↔ (Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)))
8464, 83bitri 278 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (⟨𝑓, 𝑦⟩(( Bigcup ∘ Img) ↾ (V × Limits ))𝑥 ↔ (Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)))
8526brresi 5989 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (⟨𝑓, 𝑦⟩((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))𝑥 ↔ (⟨𝑓, 𝑦⟩ ∈ (V × ran Succ) ∧ ⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup )))𝑥))
86 opelxp 5699 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (⟨𝑓, 𝑦⟩ ∈ (V × ran Succ) ↔ (𝑓 ∈ V ∧ 𝑦 ∈ ran Succ))
8712, 86mpbiran 722 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (⟨𝑓, 𝑦⟩ ∈ (V × ran Succ) ↔ 𝑦 ∈ ran Succ)
8839elrn 5885 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 ∈ ran Succ ↔ ∃𝑧 𝑧Succ𝑦)
8971, 39brsuccf 36471 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑧Succ𝑦 ↔ 𝑦 = suc 𝑧)
9089exbii 1881 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (∃𝑧 𝑧Succ𝑦 ↔ ∃𝑧 𝑦 = suc 𝑧)
9187, 88, 903bitri 300 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (⟨𝑓, 𝑦⟩ ∈ (V × ran Succ) ↔ ∃𝑧 𝑦 = suc 𝑧)
9269, 26brco 5858 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup )))𝑥 ↔ ∃𝑎(⟨𝑓, 𝑦⟩(Apply ∘ pprod( I , Bigcup ))𝑎 ∧ 𝑎FullFun𝐹𝑥))
93 vex 3461 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 𝑎 ∈ V
9469, 93brco 5858 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (⟨𝑓, 𝑦⟩(Apply ∘ pprod( I , Bigcup ))𝑎 ↔ ∃𝑧(⟨𝑓, 𝑦⟩pprod( I , Bigcup )𝑧 ∧ 𝑧Apply𝑎))
9512, 39, 71brpprod3a 36415 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (⟨𝑓, 𝑦⟩pprod( I , Bigcup )𝑧 ↔ ∃𝑎∃𝑏(𝑧 = ⟨𝑎, 𝑏⟩ ∧ 𝑓 I 𝑎 ∧ 𝑦 Bigcup 𝑏))
96 3anrot 1117 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝑧 = ⟨𝑎, 𝑏⟩ ∧ 𝑓 I 𝑎 ∧ 𝑦 Bigcup 𝑏) ↔ (𝑓 I 𝑎 ∧ 𝑦 Bigcup 𝑏 ∧ 𝑧 = ⟨𝑎, 𝑏⟩))
9793ideq 5840 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (𝑓 I 𝑎 ↔ 𝑓 = 𝑎)
98 equcom 2051 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (𝑓 = 𝑎 ↔ 𝑎 = 𝑓)
9997, 98bitri 278 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑓 I 𝑎 ↔ 𝑎 = 𝑓)
100 vex 3461 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 𝑏 ∈ V
101100brbigcup 36427 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (𝑦 Bigcup 𝑏 ↔ ∪ 𝑦 = 𝑏)
102 eqcom 2772 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (∪ 𝑦 = 𝑏 ↔ 𝑏 = ∪ 𝑦)
103101, 102bitri 278 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑦 Bigcup 𝑏 ↔ 𝑏 = ∪ 𝑦)
104 biid 264 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑧 = ⟨𝑎, 𝑏⟩ ↔ 𝑧 = ⟨𝑎, 𝑏⟩)
10599, 103, 1043anbi123i 1173 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝑓 I 𝑎 ∧ 𝑦 Bigcup 𝑏 ∧ 𝑧 = ⟨𝑎, 𝑏⟩) ↔ (𝑎 = 𝑓 ∧ 𝑏 = ∪ 𝑦 ∧ 𝑧 = ⟨𝑎, 𝑏⟩))
10696, 105bitri 278 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑧 = ⟨𝑎, 𝑏⟩ ∧ 𝑓 I 𝑎 ∧ 𝑦 Bigcup 𝑏) ↔ (𝑎 = 𝑓 ∧ 𝑏 = ∪ 𝑦 ∧ 𝑧 = ⟨𝑎, 𝑏⟩))
1071062exbii 1882 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (∃𝑎∃𝑏(𝑧 = ⟨𝑎, 𝑏⟩ ∧ 𝑓 I 𝑎 ∧ 𝑦 Bigcup 𝑏) ↔ ∃𝑎∃𝑏(𝑎 = 𝑓 ∧ 𝑏 = ∪ 𝑦 ∧ 𝑧 = ⟨𝑎, 𝑏⟩))
108 vuniex 7747 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ∪ 𝑦 ∈ V
109 opeq1 4840 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑎 = 𝑓 → ⟨𝑎, 𝑏⟩ = ⟨𝑓, 𝑏⟩)
110109eqeq2d 2776 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑎 = 𝑓 → (𝑧 = ⟨𝑎, 𝑏⟩ ↔ 𝑧 = ⟨𝑓, 𝑏⟩))
111 opeq2 4841 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑏 = ∪ 𝑦 → ⟨𝑓, 𝑏⟩ = ⟨𝑓, ∪ 𝑦⟩)
112111eqeq2d 2776 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑏 = ∪ 𝑦 → (𝑧 = ⟨𝑓, 𝑏⟩ ↔ 𝑧 = ⟨𝑓, ∪ 𝑦⟩))
11312, 108, 110, 112ceqsex2v 3508 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (∃𝑎∃𝑏(𝑎 = 𝑓 ∧ 𝑏 = ∪ 𝑦 ∧ 𝑧 = ⟨𝑎, 𝑏⟩) ↔ 𝑧 = ⟨𝑓, ∪ 𝑦⟩)
11495, 107, 1133bitri 300 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (⟨𝑓, 𝑦⟩pprod( I , Bigcup )𝑧 ↔ 𝑧 = ⟨𝑓, ∪ 𝑦⟩)
115114anbi1i 636 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((⟨𝑓, 𝑦⟩pprod( I , Bigcup )𝑧 ∧ 𝑧Apply𝑎) ↔ (𝑧 = ⟨𝑓, ∪ 𝑦⟩ ∧ 𝑧Apply𝑎))
116115exbii 1881 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (∃𝑧(⟨𝑓, 𝑦⟩pprod( I , Bigcup )𝑧 ∧ 𝑧Apply𝑎) ↔ ∃𝑧(𝑧 = ⟨𝑓, ∪ 𝑦⟩ ∧ 𝑧Apply𝑎))
117 opex 5447 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⟨𝑓, ∪ 𝑦⟩ ∈ V
118 breq1 5114 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑧 = ⟨𝑓, ∪ 𝑦⟩ → (𝑧Apply𝑎 ↔ ⟨𝑓, ∪ 𝑦⟩Apply𝑎))
119117, 118ceqsexv 3505 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (∃𝑧(𝑧 = ⟨𝑓, ∪ 𝑦⟩ ∧ 𝑧Apply𝑎) ↔ ⟨𝑓, ∪ 𝑦⟩Apply𝑎)
12012, 108, 93brapply 36467 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (⟨𝑓, ∪ 𝑦⟩Apply𝑎 ↔ 𝑎 = (𝑓‘∪ 𝑦))
121119, 120bitri 278 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (∃𝑧(𝑧 = ⟨𝑓, ∪ 𝑦⟩ ∧ 𝑧Apply𝑎) ↔ 𝑎 = (𝑓‘∪ 𝑦))
12294, 116, 1213bitri 300 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (⟨𝑓, 𝑦⟩(Apply ∘ pprod( I , Bigcup ))𝑎 ↔ 𝑎 = (𝑓‘∪ 𝑦))
12393, 26brfullfun 36479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑎FullFun𝐹𝑥 ↔ 𝑥 = (𝐹‘𝑎))
124122, 123anbi12i 640 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((⟨𝑓, 𝑦⟩(Apply ∘ pprod( I , Bigcup ))𝑎 ∧ 𝑎FullFun𝐹𝑥) ↔ (𝑎 = (𝑓‘∪ 𝑦) ∧ 𝑥 = (𝐹‘𝑎)))
125124exbii 1881 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (∃𝑎(⟨𝑓, 𝑦⟩(Apply ∘ pprod( I , Bigcup ))𝑎 ∧ 𝑎FullFun𝐹𝑥) ↔ ∃𝑎(𝑎 = (𝑓‘∪ 𝑦) ∧ 𝑥 = (𝐹‘𝑎)))
126 fvex 6898 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑓‘∪ 𝑦) ∈ V
127 fveq2 6885 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑎 = (𝑓‘∪ 𝑦) → (𝐹‘𝑎) = (𝐹‘(𝑓‘∪ 𝑦)))
128127eqeq2d 2776 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑎 = (𝑓‘∪ 𝑦) → (𝑥 = (𝐹‘𝑎) ↔ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))
129126, 128ceqsexv 3505 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (∃𝑎(𝑎 = (𝑓‘∪ 𝑦) ∧ 𝑥 = (𝐹‘𝑎)) ↔ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦)))
13092, 125, 1293bitri 300 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup )))𝑥 ↔ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦)))
13191, 130anbi12i 640 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((⟨𝑓, 𝑦⟩ ∈ (V × ran Succ) ∧ ⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup )))𝑥) ↔ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))
13285, 131bitri 278 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (⟨𝑓, 𝑦⟩((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))𝑥 ↔ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))
13384, 132orbi12i 928 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((⟨𝑓, 𝑦⟩(( Bigcup ∘ Img) ↾ (V × Limits ))𝑥 ∨ ⟨𝑓, 𝑦⟩((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))𝑥) ↔ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦)))))
13463, 133bitri 278 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (⟨𝑓, 𝑦⟩((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))𝑥 ↔ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦)))))
13562, 134orbi12i 928 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((⟨𝑓, 𝑦⟩((V × {∅}) × {∪ {𝐴}})𝑥 ∨ ⟨𝑓, 𝑦⟩((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))𝑥) ↔ ((𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}) ∨ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))))
13654, 135bitri 278 . . . . . . . . . . . . . . . . . . . . . . . 24 (⟨𝑓, 𝑦⟩(((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))𝑥 ↔ ((𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}) ∨ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))))
137 onzsl 7848 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ∈ On ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ On 𝑦 = suc 𝑧 ∨ (𝑦 ∈ V ∧ Lim 𝑦)))
138 nlim0 6425 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ¬ Lim ∅
139 limeq 6376 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 = ∅ → (Lim 𝑦 ↔ Lim ∅))
140138, 139mtbiri 330 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 = ∅ → ¬ Lim 𝑦)
141140intnanrd 495 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 = ∅ → ¬ (Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)))
142 nsuceq0 6450 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 suc 𝑧 ≠ ∅
143 neeq2 3023 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑦 = ∅ → (suc 𝑧 ≠ 𝑦 ↔ suc 𝑧 ≠ ∅))
144142, 143mpbiri 261 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑦 = ∅ → suc 𝑧 ≠ 𝑦)
145144necomd 3015 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑦 = ∅ → 𝑦 ≠ suc 𝑧)
146145neneqd 2965 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 = ∅ → ¬ 𝑦 = suc 𝑧)
147146nexdv 1969 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 = ∅ → ¬ ∃𝑧 𝑦 = suc 𝑧)
148147intnanrd 495 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 = ∅ → ¬ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))
149 ioran 999 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (¬ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦)))) ↔ (¬ (Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∧ ¬ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦)))))
150141, 148, 149sylanbrc 595 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 = ∅ → ¬ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦)))))
151 orel2 904 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (¬ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦)))) → (((𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}) ∨ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))) → (𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴})))
152150, 151syl 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 = ∅ → (((𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}) ∨ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))) → (𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴})))
153 iftrue 4495 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 = ∅ → if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) = if(𝐴 ∈ V, 𝐴, ∅))
154 unisnif 36454 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ∪ {𝐴} = if(𝐴 ∈ V, 𝐴, ∅)
155153, 154eqtr4di 2818 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 = ∅ → if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) = ∪ {𝐴})
156155eqeq2d 2776 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 = ∅ → (𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) ↔ 𝑥 = ∪ {𝐴}))
157156biimprd 251 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 = ∅ → (𝑥 = ∪ {𝐴} → 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
158157adantld 496 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 = ∅ → ((𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}) → 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
159152, 158syld 48 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 = ∅ → (((𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}) ∨ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))) → 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
160156biimpd 232 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 = ∅ → (𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) → 𝑥 = ∪ {𝐴}))
161160anc2li 565 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 = ∅ → (𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) → (𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴})))
162 orc 881 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}) → ((𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}) ∨ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))))
163161, 162syl6 36 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 = ∅ → (𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) → ((𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}) ∨ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦)))))))
164159, 163impbid 215 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 = ∅ → (((𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}) ∨ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))) ↔ 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
165 neeq1 3022 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑦 = suc 𝑧 → (𝑦 ≠ ∅ ↔ suc 𝑧 ≠ ∅))
166142, 165mpbiri 261 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 = suc 𝑧 → 𝑦 ≠ ∅)
167166neneqd 2965 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 = suc 𝑧 → ¬ 𝑦 = ∅)
168167intnanrd 495 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 = suc 𝑧 → ¬ (𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}))
169168rexlimivw 3164 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (∃𝑧 ∈ On 𝑦 = suc 𝑧 → ¬ (𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}))
170 orel1 902 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (¬ (𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}) → (((𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}) ∨ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))) → ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))))
171169, 170syl 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (∃𝑧 ∈ On 𝑦 = suc 𝑧 → (((𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}) ∨ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))) → ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))))
172 nlimsucg 7844 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑧 ∈ V → ¬ Lim suc 𝑧)
173172elv 3462 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ¬ Lim suc 𝑧
174 limeq 6376 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 = suc 𝑧 → (Lim 𝑦 ↔ Lim suc 𝑧))
175173, 174mtbiri 330 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 = suc 𝑧 → ¬ Lim 𝑦)
176175rexlimivw 3164 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (∃𝑧 ∈ On 𝑦 = suc 𝑧 → ¬ Lim 𝑦)
177176intnanrd 495 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (∃𝑧 ∈ On 𝑦 = suc 𝑧 → ¬ (Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)))
178 orel1 902 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (¬ (Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) → (((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦)))) → (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦)))))
179177, 178syl 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (∃𝑧 ∈ On 𝑦 = suc 𝑧 → (((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦)))) → (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦)))))
180142neii 2962 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ¬ suc 𝑧 = ∅
181180iffalsei 4499 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 if(suc 𝑧 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim suc 𝑧, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ suc 𝑧)))) = if(Lim suc 𝑧, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ suc 𝑧)))
182 iffalse 4498 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (¬ Lim suc 𝑧 → if(Lim suc 𝑧, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ suc 𝑧))) = (𝐹‘(𝑓‘∪ suc 𝑧)))
18371, 172, 182mp2b 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 if(Lim suc 𝑧, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ suc 𝑧))) = (𝐹‘(𝑓‘∪ suc 𝑧))
184181, 183eqtri 2788 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 if(suc 𝑧 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim suc 𝑧, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ suc 𝑧)))) = (𝐹‘(𝑓‘∪ suc 𝑧))
185 eqeq1 2769 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑦 = suc 𝑧 → (𝑦 = ∅ ↔ suc 𝑧 = ∅))
186 unieq 4885 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑦 = suc 𝑧 → ∪ 𝑦 = ∪ suc 𝑧)
187186fveq2d 6889 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑦 = suc 𝑧 → (𝑓‘∪ 𝑦) = (𝑓‘∪ suc 𝑧))
188187fveq2d 6889 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑦 = suc 𝑧 → (𝐹‘(𝑓‘∪ 𝑦)) = (𝐹‘(𝑓‘∪ suc 𝑧)))
189174, 188ifbieq2d 4516 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑦 = suc 𝑧 → if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))) = if(Lim suc 𝑧, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ suc 𝑧))))
190185, 189ifbieq2d 4516 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑦 = suc 𝑧 → if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) = if(suc 𝑧 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim suc 𝑧, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ suc 𝑧)))))
191184, 190, 1883eqtr4a 2826 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 = suc 𝑧 → if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) = (𝐹‘(𝑓‘∪ 𝑦)))
192191rexlimivw 3164 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (∃𝑧 ∈ On 𝑦 = suc 𝑧 → if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) = (𝐹‘(𝑓‘∪ 𝑦)))
193192eqeq2d 2776 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (∃𝑧 ∈ On 𝑦 = suc 𝑧 → (𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) ↔ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))
194193biimprd 251 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (∃𝑧 ∈ On 𝑦 = suc 𝑧 → (𝑥 = (𝐹‘(𝑓‘∪ 𝑦)) → 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
195194adantld 496 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (∃𝑧 ∈ On 𝑦 = suc 𝑧 → ((∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))) → 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
196171, 179, 1953syld 61 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (∃𝑧 ∈ On 𝑦 = suc 𝑧 → (((𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}) ∨ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))) → 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
197 rexex 3097 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (∃𝑧 ∈ On 𝑦 = suc 𝑧 → ∃𝑧 𝑦 = suc 𝑧)
198193biimpd 232 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (∃𝑧 ∈ On 𝑦 = suc 𝑧 → (𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) → 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))
199 olc 882 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))) → ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦)))))
200199olcd 888 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))) → ((𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}) ∨ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))))
201197, 198, 200syl6an 697 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (∃𝑧 ∈ On 𝑦 = suc 𝑧 → (𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) → ((𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}) ∨ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦)))))))
202196, 201impbid 215 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (∃𝑧 ∈ On 𝑦 = suc 𝑧 → (((𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}) ∨ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))) ↔ 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
203140con2i 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (Lim 𝑦 → ¬ 𝑦 = ∅)
204203intnanrd 495 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (Lim 𝑦 → ¬ (𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}))
205204, 170syl 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (Lim 𝑦 → (((𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}) ∨ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))) → ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))))
206175exlimiv 1963 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (∃𝑧 𝑦 = suc 𝑧 → ¬ Lim 𝑦)
207206con2i 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (Lim 𝑦 → ¬ ∃𝑧 𝑦 = suc 𝑧)
208207intnanrd 495 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (Lim 𝑦 → ¬ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))
209 orel2 904 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (¬ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))) → (((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦)))) → (Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦))))
210208, 209syl 18 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (Lim 𝑦 → (((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦)))) → (Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦))))
211203iffalsed 4500 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (Lim 𝑦 → if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) = if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))
212 iftrue 4495 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (Lim 𝑦 → if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))) = ∪ (𝑓 “ 𝑦))
213211, 212eqtrd 2800 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (Lim 𝑦 → if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) = ∪ (𝑓 “ 𝑦))
214213eqeq2d 2776 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (Lim 𝑦 → (𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) ↔ 𝑥 = ∪ (𝑓 “ 𝑦)))
215214biimprd 251 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (Lim 𝑦 → (𝑥 = ∪ (𝑓 “ 𝑦) → 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
216215adantld 496 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (Lim 𝑦 → ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) → 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
217205, 210, 2163syld 61 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (Lim 𝑦 → (((𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}) ∨ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))) → 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
218217adantl 487 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑦 ∈ V ∧ Lim 𝑦) → (((𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}) ∨ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))) → 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
219214biimpd 232 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (Lim 𝑦 → (𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) → 𝑥 = ∪ (𝑓 “ 𝑦)))
220219anc2li 565 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (Lim 𝑦 → (𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) → (Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦))))
221 orc 881 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) → ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦)))))
222221olcd 888 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) → ((𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}) ∨ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))))
223220, 222syl6 36 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (Lim 𝑦 → (𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) → ((𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}) ∨ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦)))))))
224223adantl 487 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑦 ∈ V ∧ Lim 𝑦) → (𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) → ((𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}) ∨ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦)))))))
225218, 224impbid 215 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑦 ∈ V ∧ Lim 𝑦) → (((𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}) ∨ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))) ↔ 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
226164, 202, 2253jaoi 1454 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑦 = ∅ ∨ ∃𝑧 ∈ On 𝑦 = suc 𝑧 ∨ (𝑦 ∈ V ∧ Lim 𝑦)) → (((𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}) ∨ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))) ↔ 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
227137, 226sylbi 220 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ On → (((𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}) ∨ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))) ↔ 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
228136, 227bitrid 286 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 ∈ On → (⟨𝑓, 𝑦⟩(((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))𝑥 ↔ 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
22953, 228syl 18 . . . . . . . . . . . . . . . . . . . . . 22 ((Fun 𝑓 ∧ dom 𝑓 ∈ On ∧ 𝑦 ∈ dom 𝑓) → (⟨𝑓, 𝑦⟩(((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))𝑥 ↔ 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
23026, 69brcnv 5870 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥◡Apply⟨𝑓, 𝑦⟩ ↔ ⟨𝑓, 𝑦⟩Apply𝑥)
23112, 39, 26brapply 36467 . . . . . . . . . . . . . . . . . . . . . . . 24 (⟨𝑓, 𝑦⟩Apply𝑥 ↔ 𝑥 = (𝑓‘𝑦))
232230, 231bitri 278 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥◡Apply⟨𝑓, 𝑦⟩ ↔ 𝑥 = (𝑓‘𝑦))
233232a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 ((Fun 𝑓 ∧ dom 𝑓 ∈ On ∧ 𝑦 ∈ dom 𝑓) → (𝑥◡Apply⟨𝑓, 𝑦⟩ ↔ 𝑥 = (𝑓‘𝑦)))
234229, 233anbi12d 644 . . . . . . . . . . . . . . . . . . . . 21 ((Fun 𝑓 ∧ dom 𝑓 ∈ On ∧ 𝑦 ∈ dom 𝑓) → ((⟨𝑓, 𝑦⟩(((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))𝑥 ∧ 𝑥◡Apply⟨𝑓, 𝑦⟩) ↔ (𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) ∧ 𝑥 = (𝑓‘𝑦))))
235234biancomd 469 . . . . . . . . . . . . . . . . . . . 20 ((Fun 𝑓 ∧ dom 𝑓 ∈ On ∧ 𝑦 ∈ dom 𝑓) → ((⟨𝑓, 𝑦⟩(((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))𝑥 ∧ 𝑥◡Apply⟨𝑓, 𝑦⟩) ↔ (𝑥 = (𝑓‘𝑦) ∧ 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))))
236235exbidv 1954 . . . . . . . . . . . . . . . . . . 19 ((Fun 𝑓 ∧ dom 𝑓 ∈ On ∧ 𝑦 ∈ dom 𝑓) → (∃𝑥(⟨𝑓, 𝑦⟩(((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))𝑥 ∧ 𝑥◡Apply⟨𝑓, 𝑦⟩) ↔ ∃𝑥(𝑥 = (𝑓‘𝑦) ∧ 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))))
237 df-br 5112 . . . . . . . . . . . . . . . . . . . 20 (𝑓 Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))))𝑦 ↔ ⟨𝑓, 𝑦⟩ ∈ Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))))
23869elfix 36432 . . . . . . . . . . . . . . . . . . . 20 (⟨𝑓, 𝑦⟩ ∈ Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))) ↔ ⟨𝑓, 𝑦⟩(◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))))⟨𝑓, 𝑦⟩)
23969, 69brco 5858 . . . . . . . . . . . . . . . . . . . 20 (⟨𝑓, 𝑦⟩(◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))))⟨𝑓, 𝑦⟩ ↔ ∃𝑥(⟨𝑓, 𝑦⟩(((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))𝑥 ∧ 𝑥◡Apply⟨𝑓, 𝑦⟩))
240237, 238, 2393bitri 300 . . . . . . . . . . . . . . . . . . 19 (𝑓 Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))))𝑦 ↔ ∃𝑥(⟨𝑓, 𝑦⟩(((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))𝑥 ∧ 𝑥◡Apply⟨𝑓, 𝑦⟩))
241 fvex 6898 . . . . . . . . . . . . . . . . . . . 20 (𝑓‘𝑦) ∈ V
242241eqvinc 3610 . . . . . . . . . . . . . . . . . . 19 ((𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) ↔ ∃𝑥(𝑥 = (𝑓‘𝑦) ∧ 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
243236, 240, 2423bitr4g 317 . . . . . . . . . . . . . . . . . 18 ((Fun 𝑓 ∧ dom 𝑓 ∈ On ∧ 𝑦 ∈ dom 𝑓) → (𝑓 Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))))𝑦 ↔ (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
244243notbid 321 . . . . . . . . . . . . . . . . 17 ((Fun 𝑓 ∧ dom 𝑓 ∈ On ∧ 𝑦 ∈ dom 𝑓) → (¬ 𝑓 Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))))𝑦 ↔ ¬ (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
2452443expia 1139 . . . . . . . . . . . . . . . 16 ((Fun 𝑓 ∧ dom 𝑓 ∈ On) → (𝑦 ∈ dom 𝑓 → (¬ 𝑓 Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))))𝑦 ↔ ¬ (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))))
246245pm5.32d 588 . . . . . . . . . . . . . . 15 ((Fun 𝑓 ∧ dom 𝑓 ∈ On) → ((𝑦 ∈ dom 𝑓 ∧ ¬ 𝑓 Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))))𝑦) ↔ (𝑦 ∈ dom 𝑓 ∧ ¬ (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))))
247 annim 409 . . . . . . . . . . . . . . 15 ((𝑦 ∈ dom 𝑓 ∧ ¬ (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))) ↔ ¬ (𝑦 ∈ dom 𝑓 → (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
248246, 247bitrdi 290 . . . . . . . . . . . . . 14 ((Fun 𝑓 ∧ dom 𝑓 ∈ On) → ((𝑦 ∈ dom 𝑓 ∧ ¬ 𝑓 Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))))𝑦) ↔ ¬ (𝑦 ∈ dom 𝑓 → (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))))
24951, 248bitrid 286 . . . . . . . . . . . . 13 ((Fun 𝑓 ∧ dom 𝑓 ∈ On) → (𝑓((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))))𝑦 ↔ ¬ (𝑦 ∈ dom 𝑓 → (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))))
250249exbidv 1954 . . . . . . . . . . . 12 ((Fun 𝑓 ∧ dom 𝑓 ∈ On) → (∃𝑦 𝑓((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))))𝑦 ↔ ∃𝑦 ¬ (𝑦 ∈ dom 𝑓 → (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))))
251 exnal 1860 . . . . . . . . . . . 12 (∃𝑦 ¬ (𝑦 ∈ dom 𝑓 → (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))) ↔ ¬ ∀𝑦(𝑦 ∈ dom 𝑓 → (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
252250, 251bitr2di 291 . . . . . . . . . . 11 ((Fun 𝑓 ∧ dom 𝑓 ∈ On) → (¬ ∀𝑦(𝑦 ∈ dom 𝑓 → (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))) ↔ ∃𝑦 𝑓((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))))𝑦))
25312eldm 5892 . . . . . . . . . . 11 (𝑓 ∈ dom ((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))))) ↔ ∃𝑦 𝑓((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))))𝑦)
254252, 253bitr4di 292 . . . . . . . . . 10 ((Fun 𝑓 ∧ dom 𝑓 ∈ On) → (¬ ∀𝑦(𝑦 ∈ dom 𝑓 → (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))) ↔ 𝑓 ∈ dom ((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))))))
255254con1bid 358 . . . . . . . . 9 ((Fun 𝑓 ∧ dom 𝑓 ∈ On) → (¬ 𝑓 ∈ dom ((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))))) ↔ ∀𝑦(𝑦 ∈ dom 𝑓 → (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))))
256 df-ral 3082 . . . . . . . . 9 (∀𝑦 ∈ dom 𝑓(𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) ↔ ∀𝑦(𝑦 ∈ dom 𝑓 → (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
257255, 256bitr4di 292 . . . . . . . 8 ((Fun 𝑓 ∧ dom 𝑓 ∈ On) → (¬ 𝑓 ∈ dom ((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))))) ↔ ∀𝑦 ∈ dom 𝑓(𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
258257pm5.32i 585 . . . . . . 7 (((Fun 𝑓 ∧ dom 𝑓 ∈ On) ∧ ¬ 𝑓 ∈ dom ((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))))) ↔ ((Fun 𝑓 ∧ dom 𝑓 ∈ On) ∧ ∀𝑦 ∈ dom 𝑓(𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
259 anass 474 . . . . . . 7 (((Fun 𝑓 ∧ dom 𝑓 ∈ On) ∧ ∀𝑦 ∈ dom 𝑓(𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))) ↔ (Fun 𝑓 ∧ (dom 𝑓 ∈ On ∧ ∀𝑦 ∈ dom 𝑓(𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))))
260258, 259bitri 278 . . . . . 6 (((Fun 𝑓 ∧ dom 𝑓 ∈ On) ∧ ¬ 𝑓 ∈ dom ((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))))) ↔ (Fun 𝑓 ∧ (dom 𝑓 ∈ On ∧ ∀𝑦 ∈ dom 𝑓(𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))))
26121, 37, 2603bitri 300 . . . . 5 (𝑓 ∈ (( Funs ∩ (◡Domain “ On)) ∖ dom ((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))))) ↔ (Fun 𝑓 ∧ (dom 𝑓 ∈ On ∧ ∀𝑦 ∈ dom 𝑓(𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))))
26219, 20, 2613bitr4ri 307 . . . 4 (𝑓 ∈ (( Funs ∩ (◡Domain “ On)) ∖ dom ((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))))) ↔ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))))))
263262eqabi 2900 . . 3 (( Funs ∩ (◡Domain “ On)) ∖ dom ((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))))) = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))}
264263unieqi 4886 . 2 ∪ (( Funs ∩ (◡Domain “ On)) ∖ dom ((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))))) = ∪ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))))}
2651, 264eqtr4i 2791 1 rec(𝐹, 𝐴) = ∪ (( Funs ∩ (◡Domain “ On)) ∖ dom ((◡ E ∘ Domain) ∖ Fix (◡Apply ∘ (((V × {∅}) × {∪ {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))))))
Colors of variables:    wff setvar class
This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∨ w3o 1102   ∧ w3a 1103  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2146  {cab 2743   ≠ wne 2960  ∀wral 3081  ∃wrex 3091  Vcvv 3457   ∖ cdif 3903   ∪ cun 3904   ∩ cin 3905  ∅c0 4286  ifcif 4489  {csn 4591  ⟨cop 4597  ∪ cuni 4874   class class class wbr 5111   I cid 5557   E cep 5562   × cxp 5661  ◡ccnv 5662  dom cdm 5663  ran crn 5664   ↾ cres 5665   “ cima 5666   ∘ ccom 5667  Oncon0 6364  Lim wlim 6365  suc csuc 6366  Fun wfun 6534   Fn wfn 6535  ‘cfv 6540  reccrdg 8402  pprodcpprod 36360   Bigcup cbigcup 36363   Fix cfix 36364   Limits climits 36365   Funs cfuns 36366  Imgcimg 36371  Domaincdomain 36372  Applycapply 36374  Succcsuccf 36377  FullFuncfullfn 36379
This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-rep 5240  ax-sep 5259  ax-nul 5271  ax-pow 5338  ax-pr 5406  ax-un 7742
This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-symdif 4206  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-mpt 5195  df-tr 5221  df-id 5558  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-pred 6306  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-ov 7422  df-om 7869  df-1st 7992  df-2nd 7993  df-frecs 8284  df-wrecs 8315  df-recs 8364  df-rdg 8403  df-txp 36383  df-pprod 36384  df-bigcup 36387  df-fix 36388  df-limits 36389  df-funs 36390  df-singleton 36391  df-singles 36392  df-image 36393  df-cart 36394  df-img 36395  df-domain 36396  df-cup 36398  df-succf 36401  df-apply 36402  df-funpart 36403  df-fullfun 36404
This theorem is used by: (None)
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