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Theorem dfrdg4 36443
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 36286 . 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 657 . . . . . . . 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 6539 . . . . . . . . . 10 (𝑓 Fn 𝑥 ↔ (Fun 𝑓 ∧ dom 𝑓 = 𝑥))
4 ancom 465 . . . . . . . . . 10 ((Fun 𝑓 ∧ dom 𝑓 = 𝑥) ↔ (dom 𝑓 = 𝑥 ∧ Fun 𝑓))
5 eqcom 2770 . . . . . . . . . . 11 (dom 𝑓 = 𝑥 ↔ 𝑥 = dom 𝑓)
65anbi1i 635 . . . . . . . . . 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 635 . . . . . . . 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 473 . . . . . . . 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 1878 . . . . . 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 3459 . . . . . . . 8 𝑓 ∈ V
1312dmex 7902 . . . . . . 7 dom 𝑓 ∈ V
14 eleq1 2851 . . . . . . . . 9 (𝑥 = dom 𝑓 → (𝑥 ∈ On ↔ dom 𝑓 ∈ On))
15 raleq 3320 . . . . . . . . 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 643 . . . . . . . 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 641 . . . . . . 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 3503 . . . . . 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 3090 . . . . 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 3915 . . . . . 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 3921 . . . . . . . 8 (𝑓 ∈ ( Funs ∩ (◡Domain “ On)) ↔ (𝑓 ∈ Funs ∧ 𝑓 ∈ (◡Domain “ On)))
2312elfuns 36405 . . . . . . . . 9 (𝑓 ∈ Funs ↔ Fun 𝑓)
2412elima 6067 . . . . . . . . . 10 (𝑓 ∈ (◡Domain “ On) ↔ ∃𝑥 ∈ On 𝑥◡Domain𝑓)
25 df-rex 3090 . . . . . . . . . 10 (∃𝑥 ∈ On 𝑥◡Domain𝑓 ↔ ∃𝑥(𝑥 ∈ On ∧ 𝑥◡Domain𝑓))
26 vex 3459 . . . . . . . . . . . . . . 15 𝑥 ∈ V
2726, 12brcnv 5868 . . . . . . . . . . . . . 14 (𝑥◡Domain𝑓 ↔ 𝑓Domain𝑥)
2812, 26brdomain 36423 . . . . . . . . . . . . . 14 (𝑓Domain𝑥 ↔ 𝑥 = dom 𝑓)
2927, 28bitri 278 . . . . . . . . . . . . 13 (𝑥◡Domain𝑓 ↔ 𝑥 = dom 𝑓)
3029anbi1ci 637 . . . . . . . . . . . 12 ((𝑥 ∈ On ∧ 𝑥◡Domain𝑓) ↔ (𝑥 = dom 𝑓 ∧ 𝑥 ∈ On))
3130exbii 1878 . . . . . . . . . . 11 (∃𝑥(𝑥 ∈ On ∧ 𝑥◡Domain𝑓) ↔ ∃𝑥(𝑥 = dom 𝑓 ∧ 𝑥 ∈ On))
3213, 14ceqsexv 3503 . . . . . . . . . . 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 639 . . . . . . . 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 635 . . . . . 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 5164 . . . . . . . . . . . . . . 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 3459 . . . . . . . . . . . . . . . . . 18 𝑦 ∈ V
4012, 39brco 5856 . . . . . . . . . . . . . . . . 17 (𝑓(◡ E ∘ Domain)𝑦 ↔ ∃𝑥(𝑓Domain𝑥 ∧ 𝑥◡ E 𝑦))
4128anbi1i 635 . . . . . . . . . . . . . . . . . . 19 ((𝑓Domain𝑥 ∧ 𝑥◡ E 𝑦) ↔ (𝑥 = dom 𝑓 ∧ 𝑥◡ E 𝑦))
4241exbii 1878 . . . . . . . . . . . . . . . . . 18 (∃𝑥(𝑓Domain𝑥 ∧ 𝑥◡ E 𝑦) ↔ ∃𝑥(𝑥 = dom 𝑓 ∧ 𝑥◡ E 𝑦))
43 breq1 5112 . . . . . . . . . . . . . . . . . . 19 (𝑥 = dom 𝑓 → (𝑥◡ E 𝑦 ↔ dom 𝑓◡ E 𝑦))
4413, 43ceqsexv 3503 . . . . . . . . . . . . . . . . . 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 5868 . . . . . . . . . . . . . . . . . 18 (dom 𝑓◡ E 𝑦 ↔ 𝑦 E dom 𝑓)
4713epeli 5563 . . . . . . . . . . . . . . . . . 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 635 . . . . . . . . . . . . . . 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 6385 . . . . . . . . . . . . . . . . . . . . . . . 24 ((dom 𝑓 ∈ On ∧ 𝑦 ∈ dom 𝑓) → 𝑦 ∈ On)
53523adant1 1148 . . . . . . . . . . . . . . . . . . . . . . 23 ((Fun 𝑓 ∧ dom 𝑓 ∈ On ∧ 𝑦 ∈ dom 𝑓) → 𝑦 ∈ On)
54 brun 5162 . . . . . . . . . . . . . . . . . . . . . . . . 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 5710 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (⟨𝑓, 𝑦⟩((V × {∅}) × {∪ {𝐴}})𝑥 ↔ (⟨𝑓, 𝑦⟩ ∈ (V × {∅}) ∧ 𝑥 ∈ {∪ {𝐴}}))
56 opelxp 5697 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (⟨𝑓, 𝑦⟩ ∈ (V × {∅}) ↔ (𝑓 ∈ V ∧ 𝑦 ∈ {∅}))
5712, 56mpbiran 721 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (⟨𝑓, 𝑦⟩ ∈ (V × {∅}) ↔ 𝑦 ∈ {∅})
58 velsn 4605 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
5957, 58bitri 278 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (⟨𝑓, 𝑦⟩ ∈ (V × {∅}) ↔ 𝑦 = ∅)
60 velsn 4605 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑥 ∈ {∪ {𝐴}} ↔ 𝑥 = ∪ {𝐴})
6159, 60anbi12i 639 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((⟨𝑓, 𝑦⟩ ∈ (V × {∅}) ∧ 𝑥 ∈ {∪ {𝐴}}) ↔ (𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}))
6255, 61bitri 278 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (⟨𝑓, 𝑦⟩((V × {∅}) × {∪ {𝐴}})𝑥 ↔ (𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}))
63 brun 5162 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5987 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (⟨𝑓, 𝑦⟩(( Bigcup ∘ Img) ↾ (V × Limits ))𝑥 ↔ (⟨𝑓, 𝑦⟩ ∈ (V × Limits ) ∧ ⟨𝑓, 𝑦⟩( Bigcup ∘ Img)𝑥))
65 opelxp 5697 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (⟨𝑓, 𝑦⟩ ∈ (V × Limits ) ↔ (𝑓 ∈ V ∧ 𝑦 ∈ Limits ))
6612, 65mpbiran 721 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (⟨𝑓, 𝑦⟩ ∈ (V × Limits ) ↔ 𝑦 ∈ Limits )
6739ellimits 36400 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 ∈ Limits ↔ Lim 𝑦)
6866, 67bitri 278 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (⟨𝑓, 𝑦⟩ ∈ (V × Limits ) ↔ Lim 𝑦)
69 opex 5445 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⟨𝑓, 𝑦⟩ ∈ V
7069, 26brco 5856 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (⟨𝑓, 𝑦⟩( Bigcup ∘ Img)𝑥 ↔ ∃𝑧(⟨𝑓, 𝑦⟩Img𝑧 ∧ 𝑧 Bigcup 𝑥))
71 vex 3459 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 𝑧 ∈ V
7212, 39, 71brimg 36427 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (⟨𝑓, 𝑦⟩Img𝑧 ↔ 𝑧 = (𝑓 “ 𝑦))
7326brbigcup 36388 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑧 Bigcup 𝑥 ↔ ∪ 𝑧 = 𝑥)
7472, 73anbi12i 639 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((⟨𝑓, 𝑦⟩Img𝑧 ∧ 𝑧 Bigcup 𝑥) ↔ (𝑧 = (𝑓 “ 𝑦) ∧ ∪ 𝑧 = 𝑥))
7574exbii 1878 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (∃𝑧(⟨𝑓, 𝑦⟩Img𝑧 ∧ 𝑧 Bigcup 𝑥) ↔ ∃𝑧(𝑧 = (𝑓 “ 𝑦) ∧ ∪ 𝑧 = 𝑥))
7612imaex 7907 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑓 “ 𝑦) ∈ V
77 unieq 4883 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑧 = (𝑓 “ 𝑦) → ∪ 𝑧 = ∪ (𝑓 “ 𝑦))
7877eqeq1d 2765 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑧 = (𝑓 “ 𝑦) → (∪ 𝑧 = 𝑥 ↔ ∪ (𝑓 “ 𝑦) = 𝑥))
7976, 78ceqsexv 3503 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (∃𝑧(𝑧 = (𝑓 “ 𝑦) ∧ ∪ 𝑧 = 𝑥) ↔ ∪ (𝑓 “ 𝑦) = 𝑥)
80 eqcom 2770 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 639 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5987 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5697 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (⟨𝑓, 𝑦⟩ ∈ (V × ran Succ) ↔ (𝑓 ∈ V ∧ 𝑦 ∈ ran Succ))
8712, 86mpbiran 721 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (⟨𝑓, 𝑦⟩ ∈ (V × ran Succ) ↔ 𝑦 ∈ ran Succ)
8839elrn 5883 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 ∈ ran Succ ↔ ∃𝑧 𝑧Succ𝑦)
8971, 39brsuccf 36432 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑧Succ𝑦 ↔ 𝑦 = suc 𝑧)
9089exbii 1878 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (∃𝑧 𝑧Succ𝑦 ↔ ∃𝑧 𝑦 = suc 𝑧)
9187, 88, 903bitri 300 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (⟨𝑓, 𝑦⟩ ∈ (V × ran Succ) ↔ ∃𝑧 𝑦 = suc 𝑧)
9269, 26brco 5856 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup )))𝑥 ↔ ∃𝑎(⟨𝑓, 𝑦⟩(Apply ∘ pprod( I , Bigcup ))𝑎 ∧ 𝑎FullFun𝐹𝑥))
93 vex 3459 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 𝑎 ∈ V
9469, 93brco 5856 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (⟨𝑓, 𝑦⟩(Apply ∘ pprod( I , Bigcup ))𝑎 ↔ ∃𝑧(⟨𝑓, 𝑦⟩pprod( I , Bigcup )𝑧 ∧ 𝑧Apply𝑎))
9512, 39, 71brpprod3a 36376 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5838 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (𝑓 I 𝑎 ↔ 𝑓 = 𝑎)
98 equcom 2048 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (𝑓 = 𝑎 ↔ 𝑎 = 𝑓)
9997, 98bitri 278 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑓 I 𝑎 ↔ 𝑎 = 𝑓)
100 vex 3459 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 𝑏 ∈ V
101100brbigcup 36388 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (𝑦 Bigcup 𝑏 ↔ ∪ 𝑦 = 𝑏)
102 eqcom 2770 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1879 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (∃𝑎∃𝑏(𝑧 = ⟨𝑎, 𝑏⟩ ∧ 𝑓 I 𝑎 ∧ 𝑦 Bigcup 𝑏) ↔ ∃𝑎∃𝑏(𝑎 = 𝑓 ∧ 𝑏 = ∪ 𝑦 ∧ 𝑧 = ⟨𝑎, 𝑏⟩))
108 vuniex 7737 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ∪ 𝑦 ∈ V
109 opeq1 4838 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑎 = 𝑓 → ⟨𝑎, 𝑏⟩ = ⟨𝑓, 𝑏⟩)
110109eqeq2d 2774 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑎 = 𝑓 → (𝑧 = ⟨𝑎, 𝑏⟩ ↔ 𝑧 = ⟨𝑓, 𝑏⟩))
111 opeq2 4839 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑏 = ∪ 𝑦 → ⟨𝑓, 𝑏⟩ = ⟨𝑓, ∪ 𝑦⟩)
112111eqeq2d 2774 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑏 = ∪ 𝑦 → (𝑧 = ⟨𝑓, 𝑏⟩ ↔ 𝑧 = ⟨𝑓, ∪ 𝑦⟩))
11312, 108, 110, 112ceqsex2v 3506 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 635 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((⟨𝑓, 𝑦⟩pprod( I , Bigcup )𝑧 ∧ 𝑧Apply𝑎) ↔ (𝑧 = ⟨𝑓, ∪ 𝑦⟩ ∧ 𝑧Apply𝑎))
116115exbii 1878 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (∃𝑧(⟨𝑓, 𝑦⟩pprod( I , Bigcup )𝑧 ∧ 𝑧Apply𝑎) ↔ ∃𝑧(𝑧 = ⟨𝑓, ∪ 𝑦⟩ ∧ 𝑧Apply𝑎))
117 opex 5445 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⟨𝑓, ∪ 𝑦⟩ ∈ V
118 breq1 5112 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑧 = ⟨𝑓, ∪ 𝑦⟩ → (𝑧Apply𝑎 ↔ ⟨𝑓, ∪ 𝑦⟩Apply𝑎))
119117, 118ceqsexv 3503 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (∃𝑧(𝑧 = ⟨𝑓, ∪ 𝑦⟩ ∧ 𝑧Apply𝑎) ↔ ⟨𝑓, ∪ 𝑦⟩Apply𝑎)
12012, 108, 93brapply 36428 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 36440 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑎FullFun𝐹𝑥 ↔ 𝑥 = (𝐹‘𝑎))
124122, 123anbi12i 639 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((⟨𝑓, 𝑦⟩(Apply ∘ pprod( I , Bigcup ))𝑎 ∧ 𝑎FullFun𝐹𝑥) ↔ (𝑎 = (𝑓‘∪ 𝑦) ∧ 𝑥 = (𝐹‘𝑎)))
125124exbii 1878 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (∃𝑎(⟨𝑓, 𝑦⟩(Apply ∘ pprod( I , Bigcup ))𝑎 ∧ 𝑎FullFun𝐹𝑥) ↔ ∃𝑎(𝑎 = (𝑓‘∪ 𝑦) ∧ 𝑥 = (𝐹‘𝑎)))
126 fvex 6894 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑓‘∪ 𝑦) ∈ V
127 fveq2 6881 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑎 = (𝑓‘∪ 𝑦) → (𝐹‘𝑎) = (𝐹‘(𝑓‘∪ 𝑦)))
128127eqeq2d 2774 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑎 = (𝑓‘∪ 𝑦) → (𝑥 = (𝐹‘𝑎) ↔ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))
129126, 128ceqsexv 3503 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 639 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 927 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 927 . . . . . . . . . . . . . . . . . . . . . . . . 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 7838 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ∈ On ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ On 𝑦 = suc 𝑧 ∨ (𝑦 ∈ V ∧ Lim 𝑦)))
138 nlim0 6421 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ¬ Lim ∅
139 limeq 6372 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 = ∅ → (Lim 𝑦 ↔ Lim ∅))
140138, 139mtbiri 330 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 = ∅ → ¬ Lim 𝑦)
141140intnanrd 494 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 = ∅ → ¬ (Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)))
142 nsuceq0 6446 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 suc 𝑧 ≠ ∅
143 neeq2 3021 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑦 = ∅ → (suc 𝑧 ≠ 𝑦 ↔ suc 𝑧 ≠ ∅))
144142, 143mpbiri 261 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑦 = ∅ → suc 𝑧 ≠ 𝑦)
145144necomd 3013 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑦 = ∅ → 𝑦 ≠ suc 𝑧)
146145neneqd 2963 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 = ∅ → ¬ 𝑦 = suc 𝑧)
147146nexdv 1966 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 = ∅ → ¬ ∃𝑧 𝑦 = suc 𝑧)
148147intnanrd 494 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 594 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 = ∅ → ¬ ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦)))))
151 orel2 903 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4493 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 = ∅ → if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) = if(𝐴 ∈ V, 𝐴, ∅))
154 unisnif 36415 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ∪ {𝐴} = if(𝐴 ∈ V, 𝐴, ∅)
155153, 154eqtr4di 2816 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 = ∅ → if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) = ∪ {𝐴})
156155eqeq2d 2774 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 495 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 564 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 = ∅ → (𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) → (𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴})))
162 orc 880 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3020 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑦 = suc 𝑧 → (𝑦 ≠ ∅ ↔ suc 𝑧 ≠ ∅))
166142, 165mpbiri 261 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 = suc 𝑧 → 𝑦 ≠ ∅)
167166neneqd 2963 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 = suc 𝑧 → ¬ 𝑦 = ∅)
168167intnanrd 494 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 = suc 𝑧 → ¬ (𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}))
169168rexlimivw 3162 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (∃𝑧 ∈ On 𝑦 = suc 𝑧 → ¬ (𝑦 = ∅ ∧ 𝑥 = ∪ {𝐴}))
170 orel1 901 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 7834 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑧 ∈ V → ¬ Lim suc 𝑧)
173172elv 3460 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ¬ Lim suc 𝑧
174 limeq 6372 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 = suc 𝑧 → (Lim 𝑦 ↔ Lim suc 𝑧))
175173, 174mtbiri 330 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 = suc 𝑧 → ¬ Lim 𝑦)
176175rexlimivw 3162 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (∃𝑧 ∈ On 𝑦 = suc 𝑧 → ¬ Lim 𝑦)
177176intnanrd 494 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (∃𝑧 ∈ On 𝑦 = suc 𝑧 → ¬ (Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)))
178 orel1 901 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2960 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ¬ suc 𝑧 = ∅
181180iffalsei 4497 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4496 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2786 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 if(suc 𝑧 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim suc 𝑧, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ suc 𝑧)))) = (𝐹‘(𝑓‘∪ suc 𝑧))
185 eqeq1 2767 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑦 = suc 𝑧 → (𝑦 = ∅ ↔ suc 𝑧 = ∅))
186 unieq 4883 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑦 = suc 𝑧 → ∪ 𝑦 = ∪ suc 𝑧)
187186fveq2d 6885 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑦 = suc 𝑧 → (𝑓‘∪ 𝑦) = (𝑓‘∪ suc 𝑧))
188187fveq2d 6885 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑦 = suc 𝑧 → (𝐹‘(𝑓‘∪ 𝑦)) = (𝐹‘(𝑓‘∪ suc 𝑧)))
189174, 188ifbieq2d 4514 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑦 = suc 𝑧 → if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))) = if(Lim suc 𝑧, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ suc 𝑧))))
190185, 189ifbieq2d 4514 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 2824 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 = suc 𝑧 → if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) = (𝐹‘(𝑓‘∪ 𝑦)))
192191rexlimivw 3162 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (∃𝑧 ∈ On 𝑦 = suc 𝑧 → if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) = (𝐹‘(𝑓‘∪ 𝑦)))
193192eqeq2d 2774 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 495 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3095 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 881 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))) → ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦)))))
200199olcd 887 . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 696 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 494 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1960 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (∃𝑧 𝑦 = suc 𝑧 → ¬ Lim 𝑦)
207206con2i 140 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (Lim 𝑦 → ¬ ∃𝑧 𝑦 = suc 𝑧)
208207intnanrd 494 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (Lim 𝑦 → ¬ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦))))
209 orel2 903 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4498 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 4493 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (Lim 𝑦 → if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦))) = ∪ (𝑓 “ 𝑦))
213211, 212eqtrd 2798 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (Lim 𝑦 → if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, ∪ (𝑓 “ 𝑦), (𝐹‘(𝑓‘∪ 𝑦)))) = ∪ (𝑓 “ 𝑦))
214213eqeq2d 2774 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 495 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 486 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 564 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 880 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) → ((Lim 𝑦 ∧ 𝑥 = ∪ (𝑓 “ 𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧 ∧ 𝑥 = (𝐹‘(𝑓‘∪ 𝑦)))))
222221olcd 887 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 486 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5868 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥◡Apply⟨𝑓, 𝑦⟩ ↔ ⟨𝑓, 𝑦⟩Apply𝑥)
23112, 39, 26brapply 36428 . . . . . . . . . . . . . . . . . . . . . . . 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 643 . . . . . . . . . . . . . . . . . . . . 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 468 . . . . . . . . . . . . . . . . . . . 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 1951 . . . . . . . . . . . . . . . . . . 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 5110 . . . . . . . . . . . . . . . . . . . 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 36393 . . . . . . . . . . . . . . . . . . . 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 5856 . . . . . . . . . . . . . . . . . . . 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 6894 . . . . . . . . . . . . . . . . . . . 20 (𝑓‘𝑦) ∈ V
242241eqvinc 3608 . . . . . . . . . . . . . . . . . . 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 587 . . . . . . . . . . . . . . 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 408 . . . . . . . . . . . . . . 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 1951 . . . . . . . . . . . 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 1857 . . . . . . . . . . . 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 5890 . . . . . . . . . . 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 3080 . . . . . . . . 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 584 . . . . . . 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 473 . . . . . . 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 2898 . . 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 4884 . 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 2789 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
Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   ∨ w3o 1102   ∧ w3a 1103  ∀wal 1568   = wceq 1570  ∃wex 1809   ∈ wcel 2143  {cab 2741   ≠ wne 2958  ∀wral 3079  ∃wrex 3089  Vcvv 3455   ∖ cdif 3902   ∪ cun 3903   ∩ cin 3904  ∅c0 4286  ifcif 4487  {csn 4589  ⟨cop 4595  ∪ cuni 4872   class class class wbr 5109   I cid 5555   E cep 5560   × cxp 5659  ◡ccnv 5660  dom cdm 5661  ran crn 5662   ↾ cres 5663   “ cima 5664   ∘ ccom 5665  Oncon0 6360  Lim wlim 6361  suc csuc 6362  Fun wfun 6530   Fn wfn 6531  ‘cfv 6536  reccrdg 8392  pprodcpprod 36321   Bigcup cbigcup 36324   Fix cfix 36325   Limits climits 36326   Funs cfuns 36327  Imgcimg 36332  Domaincdomain 36333  Applycapply 36335  Succcsuccf 36338  FullFuncfullfn 36340
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732
This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-symdif 4206  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-txp 36344  df-pprod 36345  df-bigcup 36348  df-fix 36349  df-limits 36350  df-funs 36351  df-singleton 36352  df-singles 36353  df-image 36354  df-cart 36355  df-img 36356  df-domain 36357  df-cup 36359  df-succf 36362  df-apply 36363  df-funpart 36364  df-fullfun 36365
This theorem is referenced by: (None)
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