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Theorem dfrdg4 33299
 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 32928 . 2 rec(𝐹, 𝐴) = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))}
2 an12 641 . . . . . . . 8 ((𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))) ↔ (𝑓 Fn 𝑥 ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))))
3 df-fn 6355 . . . . . . . . . 10 (𝑓 Fn 𝑥 ↔ (Fun 𝑓 ∧ dom 𝑓 = 𝑥))
4 ancom 461 . . . . . . . . . 10 ((Fun 𝑓 ∧ dom 𝑓 = 𝑥) ↔ (dom 𝑓 = 𝑥 ∧ Fun 𝑓))
5 eqcom 2833 . . . . . . . . . . 11 (dom 𝑓 = 𝑥𝑥 = dom 𝑓)
65anbi1i 623 . . . . . . . . . 10 ((dom 𝑓 = 𝑥 ∧ Fun 𝑓) ↔ (𝑥 = dom 𝑓 ∧ Fun 𝑓))
73, 4, 63bitri 298 . . . . . . . . 9 (𝑓 Fn 𝑥 ↔ (𝑥 = dom 𝑓 ∧ Fun 𝑓))
87anbi1i 623 . . . . . . . 8 ((𝑓 Fn 𝑥 ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))) ↔ ((𝑥 = dom 𝑓 ∧ Fun 𝑓) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))))
9 anass 469 . . . . . . . 8 (((𝑥 = dom 𝑓 ∧ Fun 𝑓) ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))) ↔ (𝑥 = dom 𝑓 ∧ (Fun 𝑓 ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))))
102, 8, 93bitri 298 . . . . . . 7 ((𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))) ↔ (𝑥 = dom 𝑓 ∧ (Fun 𝑓 ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))))
1110exbii 1841 . . . . . 6 (∃𝑥(𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))) ↔ ∃𝑥(𝑥 = dom 𝑓 ∧ (Fun 𝑓 ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))))
12 vex 3503 . . . . . . . 8 𝑓 ∈ V
1312dmex 7604 . . . . . . 7 dom 𝑓 ∈ V
14 eleq1 2905 . . . . . . . . 9 (𝑥 = dom 𝑓 → (𝑥 ∈ On ↔ dom 𝑓 ∈ On))
15 raleq 3411 . . . . . . . . 9 (𝑥 = dom 𝑓 → (∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) ↔ ∀𝑦 ∈ dom 𝑓(𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))
1614, 15anbi12d 630 . . . . . . . 8 (𝑥 = dom 𝑓 → ((𝑥 ∈ On ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))) ↔ (dom 𝑓 ∈ On ∧ ∀𝑦 ∈ dom 𝑓(𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))))
1716anbi2d 628 . . . . . . 7 (𝑥 = dom 𝑓 → ((Fun 𝑓 ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))) ↔ (Fun 𝑓 ∧ (dom 𝑓 ∈ On ∧ ∀𝑦 ∈ dom 𝑓(𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))))
1813, 17ceqsexv 3547 . . . . . 6 (∃𝑥(𝑥 = dom 𝑓 ∧ (Fun 𝑓 ∧ (𝑥 ∈ On ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))) ↔ (Fun 𝑓 ∧ (dom 𝑓 ∈ On ∧ ∀𝑦 ∈ dom 𝑓(𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))))
1911, 18bitri 276 . . . . 5 (∃𝑥(𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))) ↔ (Fun 𝑓 ∧ (dom 𝑓 ∈ On ∧ ∀𝑦 ∈ dom 𝑓(𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))))
20 df-rex 3149 . . . . 5 (∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))) ↔ ∃𝑥(𝑥 ∈ On ∧ (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))))
21 eldif 3950 . . . . . 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 4173 . . . . . . . 8 (𝑓 ∈ ( Funs ∩ (Domain “ On)) ↔ (𝑓 Funs 𝑓 ∈ (Domain “ On)))
2312elfuns 33263 . . . . . . . . 9 (𝑓 Funs ↔ Fun 𝑓)
2412elima 5932 . . . . . . . . . 10 (𝑓 ∈ (Domain “ On) ↔ ∃𝑥 ∈ On 𝑥Domain𝑓)
25 df-rex 3149 . . . . . . . . . 10 (∃𝑥 ∈ On 𝑥Domain𝑓 ↔ ∃𝑥(𝑥 ∈ On ∧ 𝑥Domain𝑓))
26 vex 3503 . . . . . . . . . . . . . . 15 𝑥 ∈ V
2726, 12brcnv 5752 . . . . . . . . . . . . . 14 (𝑥Domain𝑓𝑓Domain𝑥)
2812, 26brdomain 33281 . . . . . . . . . . . . . 14 (𝑓Domain𝑥𝑥 = dom 𝑓)
2927, 28bitri 276 . . . . . . . . . . . . 13 (𝑥Domain𝑓𝑥 = dom 𝑓)
3029anbi1ci 625 . . . . . . . . . . . 12 ((𝑥 ∈ On ∧ 𝑥Domain𝑓) ↔ (𝑥 = dom 𝑓𝑥 ∈ On))
3130exbii 1841 . . . . . . . . . . 11 (∃𝑥(𝑥 ∈ On ∧ 𝑥Domain𝑓) ↔ ∃𝑥(𝑥 = dom 𝑓𝑥 ∈ On))
3213, 14ceqsexv 3547 . . . . . . . . . . 11 (∃𝑥(𝑥 = dom 𝑓𝑥 ∈ On) ↔ dom 𝑓 ∈ On)
3331, 32bitri 276 . . . . . . . . . 10 (∃𝑥(𝑥 ∈ On ∧ 𝑥Domain𝑓) ↔ dom 𝑓 ∈ On)
3424, 25, 333bitri 298 . . . . . . . . 9 (𝑓 ∈ (Domain “ On) ↔ dom 𝑓 ∈ On)
3523, 34anbi12i 626 . . . . . . . 8 ((𝑓 Funs 𝑓 ∈ (Domain “ On)) ↔ (Fun 𝑓 ∧ dom 𝑓 ∈ On))
3622, 35bitri 276 . . . . . . 7 (𝑓 ∈ ( Funs ∩ (Domain “ On)) ↔ (Fun 𝑓 ∧ dom 𝑓 ∈ On))
3736anbi1i 623 . . . . . 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 5116 . . . . . . . . . . . . . . 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 3503 . . . . . . . . . . . . . . . . . 18 𝑦 ∈ V
4012, 39brco 5740 . . . . . . . . . . . . . . . . 17 (𝑓( E ∘ Domain)𝑦 ↔ ∃𝑥(𝑓Domain𝑥𝑥 E 𝑦))
4128anbi1i 623 . . . . . . . . . . . . . . . . . . 19 ((𝑓Domain𝑥𝑥 E 𝑦) ↔ (𝑥 = dom 𝑓𝑥 E 𝑦))
4241exbii 1841 . . . . . . . . . . . . . . . . . 18 (∃𝑥(𝑓Domain𝑥𝑥 E 𝑦) ↔ ∃𝑥(𝑥 = dom 𝑓𝑥 E 𝑦))
43 breq1 5066 . . . . . . . . . . . . . . . . . . 19 (𝑥 = dom 𝑓 → (𝑥 E 𝑦 ↔ dom 𝑓 E 𝑦))
4413, 43ceqsexv 3547 . . . . . . . . . . . . . . . . . 18 (∃𝑥(𝑥 = dom 𝑓𝑥 E 𝑦) ↔ dom 𝑓 E 𝑦)
4542, 44bitri 276 . . . . . . . . . . . . . . . . 17 (∃𝑥(𝑓Domain𝑥𝑥 E 𝑦) ↔ dom 𝑓 E 𝑦)
4613, 39brcnv 5752 . . . . . . . . . . . . . . . . . 18 (dom 𝑓 E 𝑦𝑦 E dom 𝑓)
4713epeli 5467 . . . . . . . . . . . . . . . . . 18 (𝑦 E dom 𝑓𝑦 ∈ dom 𝑓)
4846, 47bitri 276 . . . . . . . . . . . . . . . . 17 (dom 𝑓 E 𝑦𝑦 ∈ dom 𝑓)
4940, 45, 483bitri 298 . . . . . . . . . . . . . . . 16 (𝑓( E ∘ Domain)𝑦𝑦 ∈ dom 𝑓)
5049anbi1i 623 . . . . . . . . . . . . . . 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 276 . . . . . . . . . . . . . 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 6214 . . . . . . . . . . . . . . . . . . . . . . . 24 ((dom 𝑓 ∈ On ∧ 𝑦 ∈ dom 𝑓) → 𝑦 ∈ On)
53523adant1 1124 . . . . . . . . . . . . . . . . . . . . . . 23 ((Fun 𝑓 ∧ dom 𝑓 ∈ On ∧ 𝑦 ∈ dom 𝑓) → 𝑦 ∈ On)
54 brun 5114 . . . . . . . . . . . . . . . . . . . . . . . . 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 5600 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (⟨𝑓, 𝑦⟩((V × {∅}) × { {𝐴}})𝑥 ↔ (⟨𝑓, 𝑦⟩ ∈ (V × {∅}) ∧ 𝑥 ∈ { {𝐴}}))
56 opelxp 5590 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (⟨𝑓, 𝑦⟩ ∈ (V × {∅}) ↔ (𝑓 ∈ V ∧ 𝑦 ∈ {∅}))
5712, 56mpbiran 705 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (⟨𝑓, 𝑦⟩ ∈ (V × {∅}) ↔ 𝑦 ∈ {∅})
58 velsn 4580 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 ∈ {∅} ↔ 𝑦 = ∅)
5957, 58bitri 276 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (⟨𝑓, 𝑦⟩ ∈ (V × {∅}) ↔ 𝑦 = ∅)
60 velsn 4580 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑥 ∈ { {𝐴}} ↔ 𝑥 = {𝐴})
6159, 60anbi12i 626 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((⟨𝑓, 𝑦⟩ ∈ (V × {∅}) ∧ 𝑥 ∈ { {𝐴}}) ↔ (𝑦 = ∅ ∧ 𝑥 = {𝐴}))
6255, 61bitri 276 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (⟨𝑓, 𝑦⟩((V × {∅}) × { {𝐴}})𝑥 ↔ (𝑦 = ∅ ∧ 𝑥 = {𝐴}))
63 brun 5114 . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5861 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (⟨𝑓, 𝑦⟩(( Bigcup ∘ Img) ↾ (V × Limits ))𝑥 ↔ (⟨𝑓, 𝑦⟩ ∈ (V × Limits ) ∧ ⟨𝑓, 𝑦⟩( Bigcup ∘ Img)𝑥))
65 opelxp 5590 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (⟨𝑓, 𝑦⟩ ∈ (V × Limits ) ↔ (𝑓 ∈ V ∧ 𝑦 Limits ))
6612, 65mpbiran 705 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (⟨𝑓, 𝑦⟩ ∈ (V × Limits ) ↔ 𝑦 Limits )
6739ellimits 33258 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 Limits ↔ Lim 𝑦)
6866, 67bitri 276 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (⟨𝑓, 𝑦⟩ ∈ (V × Limits ) ↔ Lim 𝑦)
69 opex 5353 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 𝑓, 𝑦⟩ ∈ V
7069, 26brco 5740 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (⟨𝑓, 𝑦⟩( Bigcup ∘ Img)𝑥 ↔ ∃𝑧(⟨𝑓, 𝑦⟩Img𝑧𝑧 Bigcup 𝑥))
71 vex 3503 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 𝑧 ∈ V
7212, 39, 71brimg 33285 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (⟨𝑓, 𝑦⟩Img𝑧𝑧 = (𝑓𝑦))
7326brbigcup 33246 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑧 Bigcup 𝑥 𝑧 = 𝑥)
7472, 73anbi12i 626 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((⟨𝑓, 𝑦⟩Img𝑧𝑧 Bigcup 𝑥) ↔ (𝑧 = (𝑓𝑦) ∧ 𝑧 = 𝑥))
7574exbii 1841 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (∃𝑧(⟨𝑓, 𝑦⟩Img𝑧𝑧 Bigcup 𝑥) ↔ ∃𝑧(𝑧 = (𝑓𝑦) ∧ 𝑧 = 𝑥))
7612imaex 7609 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑓𝑦) ∈ V
77 unieq 4845 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑧 = (𝑓𝑦) → 𝑧 = (𝑓𝑦))
7877eqeq1d 2828 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑧 = (𝑓𝑦) → ( 𝑧 = 𝑥 (𝑓𝑦) = 𝑥))
7976, 78ceqsexv 3547 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (∃𝑧(𝑧 = (𝑓𝑦) ∧ 𝑧 = 𝑥) ↔ (𝑓𝑦) = 𝑥)
80 eqcom 2833 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ( (𝑓𝑦) = 𝑥𝑥 = (𝑓𝑦))
8179, 80bitri 276 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (∃𝑧(𝑧 = (𝑓𝑦) ∧ 𝑧 = 𝑥) ↔ 𝑥 = (𝑓𝑦))
8270, 75, 813bitri 298 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (⟨𝑓, 𝑦⟩( Bigcup ∘ Img)𝑥𝑥 = (𝑓𝑦))
8368, 82anbi12i 626 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((⟨𝑓, 𝑦⟩ ∈ (V × Limits ) ∧ ⟨𝑓, 𝑦⟩( Bigcup ∘ Img)𝑥) ↔ (Lim 𝑦𝑥 = (𝑓𝑦)))
8464, 83bitri 276 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (⟨𝑓, 𝑦⟩(( Bigcup ∘ Img) ↾ (V × Limits ))𝑥 ↔ (Lim 𝑦𝑥 = (𝑓𝑦)))
8526brresi 5861 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 5590 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (⟨𝑓, 𝑦⟩ ∈ (V × ran Succ) ↔ (𝑓 ∈ V ∧ 𝑦 ∈ ran Succ))
8712, 86mpbiran 705 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (⟨𝑓, 𝑦⟩ ∈ (V × ran Succ) ↔ 𝑦 ∈ ran Succ)
8839elrn 5821 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 ∈ ran Succ ↔ ∃𝑧 𝑧Succ𝑦)
8971, 39brsuccf 33289 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑧Succ𝑦𝑦 = suc 𝑧)
9089exbii 1841 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (∃𝑧 𝑧Succ𝑦 ↔ ∃𝑧 𝑦 = suc 𝑧)
9187, 88, 903bitri 298 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (⟨𝑓, 𝑦⟩ ∈ (V × ran Succ) ↔ ∃𝑧 𝑦 = suc 𝑧)
9269, 26brco 5740 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup )))𝑥 ↔ ∃𝑎(⟨𝑓, 𝑦⟩(Apply ∘ pprod( I , Bigcup ))𝑎𝑎FullFun𝐹𝑥))
93 vex 3503 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 𝑎 ∈ V
9469, 93brco 5740 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (⟨𝑓, 𝑦⟩(Apply ∘ pprod( I , Bigcup ))𝑎 ↔ ∃𝑧(⟨𝑓, 𝑦⟩pprod( I , Bigcup )𝑧𝑧Apply𝑎))
9512, 39, 71brpprod3a 33234 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (⟨𝑓, 𝑦⟩pprod( I , Bigcup )𝑧 ↔ ∃𝑎𝑏(𝑧 = ⟨𝑎, 𝑏⟩ ∧ 𝑓 I 𝑎𝑦 Bigcup 𝑏))
96 3anrot 1094 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝑧 = ⟨𝑎, 𝑏⟩ ∧ 𝑓 I 𝑎𝑦 Bigcup 𝑏) ↔ (𝑓 I 𝑎𝑦 Bigcup 𝑏𝑧 = ⟨𝑎, 𝑏⟩))
9793ideq 5722 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (𝑓 I 𝑎𝑓 = 𝑎)
98 equcom 2018 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (𝑓 = 𝑎𝑎 = 𝑓)
9997, 98bitri 276 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑓 I 𝑎𝑎 = 𝑓)
100 vex 3503 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 𝑏 ∈ V
101100brbigcup 33246 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 (𝑦 Bigcup 𝑏 𝑦 = 𝑏)
102 eqcom 2833 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 ( 𝑦 = 𝑏𝑏 = 𝑦)
103101, 102bitri 276 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑦 Bigcup 𝑏𝑏 = 𝑦)
104 biid 262 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 (𝑧 = ⟨𝑎, 𝑏⟩ ↔ 𝑧 = ⟨𝑎, 𝑏⟩)
10599, 103, 1043anbi123i 1149 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 ((𝑓 I 𝑎𝑦 Bigcup 𝑏𝑧 = ⟨𝑎, 𝑏⟩) ↔ (𝑎 = 𝑓𝑏 = 𝑦𝑧 = ⟨𝑎, 𝑏⟩))
10696, 105bitri 276 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ((𝑧 = ⟨𝑎, 𝑏⟩ ∧ 𝑓 I 𝑎𝑦 Bigcup 𝑏) ↔ (𝑎 = 𝑓𝑏 = 𝑦𝑧 = ⟨𝑎, 𝑏⟩))
1071062exbii 1842 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (∃𝑎𝑏(𝑧 = ⟨𝑎, 𝑏⟩ ∧ 𝑓 I 𝑎𝑦 Bigcup 𝑏) ↔ ∃𝑎𝑏(𝑎 = 𝑓𝑏 = 𝑦𝑧 = ⟨𝑎, 𝑏⟩))
108 vuniex 7456 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 𝑦 ∈ V
109 opeq1 4802 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑎 = 𝑓 → ⟨𝑎, 𝑏⟩ = ⟨𝑓, 𝑏⟩)
110109eqeq2d 2837 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑎 = 𝑓 → (𝑧 = ⟨𝑎, 𝑏⟩ ↔ 𝑧 = ⟨𝑓, 𝑏⟩))
111 opeq2 4803 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 (𝑏 = 𝑦 → ⟨𝑓, 𝑏⟩ = ⟨𝑓, 𝑦⟩)
112111eqeq2d 2837 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 (𝑏 = 𝑦 → (𝑧 = ⟨𝑓, 𝑏⟩ ↔ 𝑧 = ⟨𝑓, 𝑦⟩))
11312, 108, 110, 112ceqsex2v 3550 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (∃𝑎𝑏(𝑎 = 𝑓𝑏 = 𝑦𝑧 = ⟨𝑎, 𝑏⟩) ↔ 𝑧 = ⟨𝑓, 𝑦⟩)
11495, 107, 1133bitri 298 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (⟨𝑓, 𝑦⟩pprod( I , Bigcup )𝑧𝑧 = ⟨𝑓, 𝑦⟩)
115114anbi1i 623 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ((⟨𝑓, 𝑦⟩pprod( I , Bigcup )𝑧𝑧Apply𝑎) ↔ (𝑧 = ⟨𝑓, 𝑦⟩ ∧ 𝑧Apply𝑎))
116115exbii 1841 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (∃𝑧(⟨𝑓, 𝑦⟩pprod( I , Bigcup )𝑧𝑧Apply𝑎) ↔ ∃𝑧(𝑧 = ⟨𝑓, 𝑦⟩ ∧ 𝑧Apply𝑎))
117 opex 5353 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 𝑓, 𝑦⟩ ∈ V
118 breq1 5066 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑧 = ⟨𝑓, 𝑦⟩ → (𝑧Apply𝑎 ↔ ⟨𝑓, 𝑦⟩Apply𝑎))
119117, 118ceqsexv 3547 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (∃𝑧(𝑧 = ⟨𝑓, 𝑦⟩ ∧ 𝑧Apply𝑎) ↔ ⟨𝑓, 𝑦⟩Apply𝑎)
12012, 108, 93brapply 33286 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (⟨𝑓, 𝑦⟩Apply𝑎𝑎 = (𝑓 𝑦))
121119, 120bitri 276 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (∃𝑧(𝑧 = ⟨𝑓, 𝑦⟩ ∧ 𝑧Apply𝑎) ↔ 𝑎 = (𝑓 𝑦))
12294, 116, 1213bitri 298 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (⟨𝑓, 𝑦⟩(Apply ∘ pprod( I , Bigcup ))𝑎𝑎 = (𝑓 𝑦))
12393, 26brfullfun 33296 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑎FullFun𝐹𝑥𝑥 = (𝐹𝑎))
124122, 123anbi12i 626 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((⟨𝑓, 𝑦⟩(Apply ∘ pprod( I , Bigcup ))𝑎𝑎FullFun𝐹𝑥) ↔ (𝑎 = (𝑓 𝑦) ∧ 𝑥 = (𝐹𝑎)))
125124exbii 1841 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (∃𝑎(⟨𝑓, 𝑦⟩(Apply ∘ pprod( I , Bigcup ))𝑎𝑎FullFun𝐹𝑥) ↔ ∃𝑎(𝑎 = (𝑓 𝑦) ∧ 𝑥 = (𝐹𝑎)))
126 fvex 6680 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑓 𝑦) ∈ V
127 fveq2 6667 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑎 = (𝑓 𝑦) → (𝐹𝑎) = (𝐹‘(𝑓 𝑦)))
128127eqeq2d 2837 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑎 = (𝑓 𝑦) → (𝑥 = (𝐹𝑎) ↔ 𝑥 = (𝐹‘(𝑓 𝑦))))
129126, 128ceqsexv 3547 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (∃𝑎(𝑎 = (𝑓 𝑦) ∧ 𝑥 = (𝐹𝑎)) ↔ 𝑥 = (𝐹‘(𝑓 𝑦)))
13092, 125, 1293bitri 298 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup )))𝑥𝑥 = (𝐹‘(𝑓 𝑦)))
13191, 130anbi12i 626 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((⟨𝑓, 𝑦⟩ ∈ (V × ran Succ) ∧ ⟨𝑓, 𝑦⟩(FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup )))𝑥) ↔ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦))))
13285, 131bitri 276 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (⟨𝑓, 𝑦⟩((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))𝑥 ↔ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦))))
13384, 132orbi12i 910 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((⟨𝑓, 𝑦⟩(( Bigcup ∘ Img) ↾ (V × Limits ))𝑥 ∨ ⟨𝑓, 𝑦⟩((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))𝑥) ↔ ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦)))))
13463, 133bitri 276 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (⟨𝑓, 𝑦⟩((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ)))𝑥 ↔ ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦)))))
13562, 134orbi12i 910 . . . . . . . . . . . . . . . . . . . . . . . . 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 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦))))))
13654, 135bitri 276 . . . . . . . . . . . . . . . . . . . . . . . 24 (⟨𝑓, 𝑦⟩(((V × {∅}) × { {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))𝑥 ↔ ((𝑦 = ∅ ∧ 𝑥 = {𝐴}) ∨ ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦))))))
137 onzsl 7549 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑦 ∈ On ↔ (𝑦 = ∅ ∨ ∃𝑧 ∈ On 𝑦 = suc 𝑧 ∨ (𝑦 ∈ V ∧ Lim 𝑦)))
138 nlim0 6247 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ¬ Lim ∅
139 limeq 6201 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 = ∅ → (Lim 𝑦 ↔ Lim ∅))
140138, 139mtbiri 328 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 = ∅ → ¬ Lim 𝑦)
141140intnanrd 490 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 = ∅ → ¬ (Lim 𝑦𝑥 = (𝑓𝑦)))
142 nsuceq0 6269 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 suc 𝑧 ≠ ∅
143 neeq2 3084 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑦 = ∅ → (suc 𝑧𝑦 ↔ suc 𝑧 ≠ ∅))
144142, 143mpbiri 259 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑦 = ∅ → suc 𝑧𝑦)
145144necomd 3076 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑦 = ∅ → 𝑦 ≠ suc 𝑧)
146145neneqd 3026 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 = ∅ → ¬ 𝑦 = suc 𝑧)
147146nexdv 1930 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 = ∅ → ¬ ∃𝑧 𝑦 = suc 𝑧)
148147intnanrd 490 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 = ∅ → ¬ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦))))
149 ioran 979 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (¬ ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦)))) ↔ (¬ (Lim 𝑦𝑥 = (𝑓𝑦)) ∧ ¬ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦)))))
150141, 148, 149sylanbrc 583 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 = ∅ → ¬ ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦)))))
151 orel2 886 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (¬ ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦)))) → (((𝑦 = ∅ ∧ 𝑥 = {𝐴}) ∨ ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦))))) → (𝑦 = ∅ ∧ 𝑥 = {𝐴})))
152150, 151syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 = ∅ → (((𝑦 = ∅ ∧ 𝑥 = {𝐴}) ∨ ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦))))) → (𝑦 = ∅ ∧ 𝑥 = {𝐴})))
153 iftrue 4476 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 = ∅ → if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) = if(𝐴 ∈ V, 𝐴, ∅))
154 unisnif 33273 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 {𝐴} = if(𝐴 ∈ V, 𝐴, ∅)
155153, 154syl6eqr 2879 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 = ∅ → if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) = {𝐴})
156155eqeq2d 2837 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 = ∅ → (𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) ↔ 𝑥 = {𝐴}))
157156biimprd 249 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 = ∅ → (𝑥 = {𝐴} → 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))
158157adantld 491 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 = ∅ → ((𝑦 = ∅ ∧ 𝑥 = {𝐴}) → 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))
159152, 158syld 47 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 = ∅ → (((𝑦 = ∅ ∧ 𝑥 = {𝐴}) ∨ ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦))))) → 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))
160156biimpd 230 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑦 = ∅ → (𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) → 𝑥 = {𝐴}))
161160anc2li 556 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑦 = ∅ → (𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) → (𝑦 = ∅ ∧ 𝑥 = {𝐴})))
162 orc 863 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((𝑦 = ∅ ∧ 𝑥 = {𝐴}) → ((𝑦 = ∅ ∧ 𝑥 = {𝐴}) ∨ ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦))))))
163161, 162syl6 35 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑦 = ∅ → (𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) → ((𝑦 = ∅ ∧ 𝑥 = {𝐴}) ∨ ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦)))))))
164159, 163impbid 213 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 = ∅ → (((𝑦 = ∅ ∧ 𝑥 = {𝐴}) ∨ ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦))))) ↔ 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))
165 neeq1 3083 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑦 = suc 𝑧 → (𝑦 ≠ ∅ ↔ suc 𝑧 ≠ ∅))
166142, 165mpbiri 259 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 = suc 𝑧𝑦 ≠ ∅)
167166neneqd 3026 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 = suc 𝑧 → ¬ 𝑦 = ∅)
168167intnanrd 490 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑦 = suc 𝑧 → ¬ (𝑦 = ∅ ∧ 𝑥 = {𝐴}))
169168rexlimivw 3287 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (∃𝑧 ∈ On 𝑦 = suc 𝑧 → ¬ (𝑦 = ∅ ∧ 𝑥 = {𝐴}))
170 orel1 884 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (¬ (𝑦 = ∅ ∧ 𝑥 = {𝐴}) → (((𝑦 = ∅ ∧ 𝑥 = {𝐴}) ∨ ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦))))) → ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦))))))
171169, 170syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (∃𝑧 ∈ On 𝑦 = suc 𝑧 → (((𝑦 = ∅ ∧ 𝑥 = {𝐴}) ∨ ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦))))) → ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦))))))
172 nlimsucg 7545 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑧 ∈ V → ¬ Lim suc 𝑧)
173172elv 3505 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ¬ Lim suc 𝑧
174 limeq 6201 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 = suc 𝑧 → (Lim 𝑦 ↔ Lim suc 𝑧))
175173, 174mtbiri 328 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑦 = suc 𝑧 → ¬ Lim 𝑦)
176175rexlimivw 3287 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (∃𝑧 ∈ On 𝑦 = suc 𝑧 → ¬ Lim 𝑦)
177176intnanrd 490 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (∃𝑧 ∈ On 𝑦 = suc 𝑧 → ¬ (Lim 𝑦𝑥 = (𝑓𝑦)))
178 orel1 884 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (¬ (Lim 𝑦𝑥 = (𝑓𝑦)) → (((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦)))) → (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦)))))
179177, 178syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (∃𝑧 ∈ On 𝑦 = suc 𝑧 → (((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦)))) → (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦)))))
180142neii 3023 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ¬ suc 𝑧 = ∅
181180iffalsei 4480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 if(suc 𝑧 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim suc 𝑧, (𝑓𝑦), (𝐹‘(𝑓 suc 𝑧)))) = if(Lim suc 𝑧, (𝑓𝑦), (𝐹‘(𝑓 suc 𝑧)))
182 iffalse 4479 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (¬ Lim suc 𝑧 → if(Lim suc 𝑧, (𝑓𝑦), (𝐹‘(𝑓 suc 𝑧))) = (𝐹‘(𝑓 suc 𝑧)))
18371, 172, 182mp2b 10 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 if(Lim suc 𝑧, (𝑓𝑦), (𝐹‘(𝑓 suc 𝑧))) = (𝐹‘(𝑓 suc 𝑧))
184181, 183eqtri 2849 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 if(suc 𝑧 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim suc 𝑧, (𝑓𝑦), (𝐹‘(𝑓 suc 𝑧)))) = (𝐹‘(𝑓 suc 𝑧))
185 eqeq1 2830 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑦 = suc 𝑧 → (𝑦 = ∅ ↔ suc 𝑧 = ∅))
186 unieq 4845 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 (𝑦 = suc 𝑧 𝑦 = suc 𝑧)
187186fveq2d 6671 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 (𝑦 = suc 𝑧 → (𝑓 𝑦) = (𝑓 suc 𝑧))
188187fveq2d 6671 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 (𝑦 = suc 𝑧 → (𝐹‘(𝑓 𝑦)) = (𝐹‘(𝑓 suc 𝑧)))
189174, 188ifbieq2d 4495 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑦 = suc 𝑧 → if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))) = if(Lim suc 𝑧, (𝑓𝑦), (𝐹‘(𝑓 suc 𝑧))))
190185, 189ifbieq2d 4495 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑦 = suc 𝑧 → if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) = if(suc 𝑧 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim suc 𝑧, (𝑓𝑦), (𝐹‘(𝑓 suc 𝑧)))))
191184, 190, 1883eqtr4a 2887 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑦 = suc 𝑧 → if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) = (𝐹‘(𝑓 𝑦)))
192191rexlimivw 3287 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (∃𝑧 ∈ On 𝑦 = suc 𝑧 → if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) = (𝐹‘(𝑓 𝑦)))
193192eqeq2d 2837 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (∃𝑧 ∈ On 𝑦 = suc 𝑧 → (𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) ↔ 𝑥 = (𝐹‘(𝑓 𝑦))))
194193biimprd 249 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (∃𝑧 ∈ On 𝑦 = suc 𝑧 → (𝑥 = (𝐹‘(𝑓 𝑦)) → 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))
195194adantld 491 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (∃𝑧 ∈ On 𝑦 = suc 𝑧 → ((∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦))) → 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))
196171, 179, 1953syld 60 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (∃𝑧 ∈ On 𝑦 = suc 𝑧 → (((𝑦 = ∅ ∧ 𝑥 = {𝐴}) ∨ ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦))))) → 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))
197 rexex 3245 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (∃𝑧 ∈ On 𝑦 = suc 𝑧 → ∃𝑧 𝑦 = suc 𝑧)
198193biimpd 230 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (∃𝑧 ∈ On 𝑦 = suc 𝑧 → (𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) → 𝑥 = (𝐹‘(𝑓 𝑦))))
199 olc 864 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦))) → ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦)))))
200199olcd 872 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦))) → ((𝑦 = ∅ ∧ 𝑥 = {𝐴}) ∨ ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦))))))
201197, 198, 200syl6an 680 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (∃𝑧 ∈ On 𝑦 = suc 𝑧 → (𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) → ((𝑦 = ∅ ∧ 𝑥 = {𝐴}) ∨ ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦)))))))
202196, 201impbid 213 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (∃𝑧 ∈ On 𝑦 = suc 𝑧 → (((𝑦 = ∅ ∧ 𝑥 = {𝐴}) ∨ ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦))))) ↔ 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))
203140con2i 141 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (Lim 𝑦 → ¬ 𝑦 = ∅)
204203intnanrd 490 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (Lim 𝑦 → ¬ (𝑦 = ∅ ∧ 𝑥 = {𝐴}))
205204, 170syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (Lim 𝑦 → (((𝑦 = ∅ ∧ 𝑥 = {𝐴}) ∨ ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦))))) → ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦))))))
206175exlimiv 1924 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (∃𝑧 𝑦 = suc 𝑧 → ¬ Lim 𝑦)
207206con2i 141 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (Lim 𝑦 → ¬ ∃𝑧 𝑦 = suc 𝑧)
208207intnanrd 490 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (Lim 𝑦 → ¬ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦))))
209 orel2 886 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (¬ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦))) → (((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦)))) → (Lim 𝑦𝑥 = (𝑓𝑦))))
210208, 209syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (Lim 𝑦 → (((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦)))) → (Lim 𝑦𝑥 = (𝑓𝑦))))
211203iffalsed 4481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (Lim 𝑦 → if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) = if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))
212 iftrue 4476 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (Lim 𝑦 → if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))) = (𝑓𝑦))
213211, 212eqtrd 2861 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (Lim 𝑦 → if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) = (𝑓𝑦))
214213eqeq2d 2837 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (Lim 𝑦 → (𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) ↔ 𝑥 = (𝑓𝑦)))
215214biimprd 249 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (Lim 𝑦 → (𝑥 = (𝑓𝑦) → 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))
216215adantld 491 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (Lim 𝑦 → ((Lim 𝑦𝑥 = (𝑓𝑦)) → 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))
217205, 210, 2163syld 60 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (Lim 𝑦 → (((𝑦 = ∅ ∧ 𝑥 = {𝐴}) ∨ ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦))))) → 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))
218217adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑦 ∈ V ∧ Lim 𝑦) → (((𝑦 = ∅ ∧ 𝑥 = {𝐴}) ∨ ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦))))) → 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))
219214biimpd 230 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (Lim 𝑦 → (𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) → 𝑥 = (𝑓𝑦)))
220219anc2li 556 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (Lim 𝑦 → (𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) → (Lim 𝑦𝑥 = (𝑓𝑦))))
221 orc 863 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((Lim 𝑦𝑥 = (𝑓𝑦)) → ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦)))))
222221olcd 872 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((Lim 𝑦𝑥 = (𝑓𝑦)) → ((𝑦 = ∅ ∧ 𝑥 = {𝐴}) ∨ ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦))))))
223220, 222syl6 35 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (Lim 𝑦 → (𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) → ((𝑦 = ∅ ∧ 𝑥 = {𝐴}) ∨ ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦)))))))
224223adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝑦 ∈ V ∧ Lim 𝑦) → (𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) → ((𝑦 = ∅ ∧ 𝑥 = {𝐴}) ∨ ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦)))))))
225218, 224impbid 213 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝑦 ∈ V ∧ Lim 𝑦) → (((𝑦 = ∅ ∧ 𝑥 = {𝐴}) ∨ ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦))))) ↔ 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))
226164, 202, 2253jaoi 1421 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑦 = ∅ ∨ ∃𝑧 ∈ On 𝑦 = suc 𝑧 ∨ (𝑦 ∈ V ∧ Lim 𝑦)) → (((𝑦 = ∅ ∧ 𝑥 = {𝐴}) ∨ ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦))))) ↔ 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))
227137, 226sylbi 218 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑦 ∈ On → (((𝑦 = ∅ ∧ 𝑥 = {𝐴}) ∨ ((Lim 𝑦𝑥 = (𝑓𝑦)) ∨ (∃𝑧 𝑦 = suc 𝑧𝑥 = (𝐹‘(𝑓 𝑦))))) ↔ 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))
228136, 227syl5bb 284 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑦 ∈ On → (⟨𝑓, 𝑦⟩(((V × {∅}) × { {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))𝑥𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))
22953, 228syl 17 . . . . . . . . . . . . . . . . . . . . . 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))))𝑥𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))
23026, 69brcnv 5752 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥Apply⟨𝑓, 𝑦⟩ ↔ ⟨𝑓, 𝑦⟩Apply𝑥)
23112, 39, 26brapply 33286 . . . . . . . . . . . . . . . . . . . . . . . 24 (⟨𝑓, 𝑦⟩Apply𝑥𝑥 = (𝑓𝑦))
232230, 231bitri 276 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥Apply⟨𝑓, 𝑦⟩ ↔ 𝑥 = (𝑓𝑦))
233232a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 ((Fun 𝑓 ∧ dom 𝑓 ∈ On ∧ 𝑦 ∈ dom 𝑓) → (𝑥Apply⟨𝑓, 𝑦⟩ ↔ 𝑥 = (𝑓𝑦)))
234229, 233anbi12d 630 . . . . . . . . . . . . . . . . . . . . 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))))𝑥𝑥Apply⟨𝑓, 𝑦⟩) ↔ (𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) ∧ 𝑥 = (𝑓𝑦))))
235234biancomd 464 . . . . . . . . . . . . . . . . . . . 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))))𝑥𝑥Apply⟨𝑓, 𝑦⟩) ↔ (𝑥 = (𝑓𝑦) ∧ 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))))
236235exbidv 1915 . . . . . . . . . . . . . . . . . . 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))))𝑥𝑥Apply⟨𝑓, 𝑦⟩) ↔ ∃𝑥(𝑥 = (𝑓𝑦) ∧ 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))))
237 df-br 5064 . . . . . . . . . . . . . . . . . . . 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 33251 . . . . . . . . . . . . . . . . . . . 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 5740 . . . . . . . . . . . . . . . . . . . 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))))𝑥𝑥Apply⟨𝑓, 𝑦⟩))
240237, 238, 2393bitri 298 . . . . . . . . . . . . . . . . . . 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))))𝑥𝑥Apply⟨𝑓, 𝑦⟩))
241 fvex 6680 . . . . . . . . . . . . . . . . . . . 20 (𝑓𝑦) ∈ V
242241eqvinc 3646 . . . . . . . . . . . . . . . . . . 19 ((𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) ↔ ∃𝑥(𝑥 = (𝑓𝑦) ∧ 𝑥 = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))
243236, 240, 2423bitr4g 315 . . . . . . . . . . . . . . . . . 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 ↔ (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))
244243notbid 319 . . . . . . . . . . . . . . . . 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 ↔ ¬ (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))
2452443expia 1115 . . . . . . . . . . . . . . . 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 ↔ ¬ (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))))
246245pm5.32d 577 . . . . . . . . . . . . . . 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 ∧ ¬ (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))))
247 annim 404 . . . . . . . . . . . . . . 15 ((𝑦 ∈ dom 𝑓 ∧ ¬ (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))) ↔ ¬ (𝑦 ∈ dom 𝑓 → (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))
248246, 247syl6bb 288 . . . . . . . . . . . . . 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 → (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))))
24951, 248syl5bb 284 . . . . . . . . . . . . 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 → (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))))
250249exbidv 1915 . . . . . . . . . . . 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 → (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))))
251 exnal 1820 . . . . . . . . . . . 12 (∃𝑦 ¬ (𝑦 ∈ dom 𝑓 → (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))) ↔ ¬ ∀𝑦(𝑦 ∈ dom 𝑓 → (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))
252250, 251syl6rbb 289 . . . . . . . . . . 11 ((Fun 𝑓 ∧ dom 𝑓 ∈ On) → (¬ ∀𝑦(𝑦 ∈ dom 𝑓 → (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (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 5768 . . . . . . . . . . 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, 253syl6bbr 290 . . . . . . . . . 10 ((Fun 𝑓 ∧ dom 𝑓 ∈ On) → (¬ ∀𝑦(𝑦 ∈ dom 𝑓 → (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))) ↔ 𝑓 ∈ dom (( E ∘ Domain) ∖ Fix (Apply ∘ (((V × {∅}) × { {𝐴}}) ∪ ((( Bigcup ∘ Img) ↾ (V × Limits )) ∪ ((FullFun𝐹 ∘ (Apply ∘ pprod( I , Bigcup ))) ↾ (V × ran Succ))))))))
255254con1bid 357 . . . . . . . . 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 → (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))))
256 df-ral 3148 . . . . . . . . 9 (∀𝑦 ∈ dom 𝑓(𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))) ↔ ∀𝑦(𝑦 ∈ dom 𝑓 → (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))
257255, 256syl6bbr 290 . . . . . . . 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(𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))
258257pm5.32i 575 . . . . . . 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(𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))
259 anass 469 . . . . . . 7 (((Fun 𝑓 ∧ dom 𝑓 ∈ On) ∧ ∀𝑦 ∈ dom 𝑓(𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))) ↔ (Fun 𝑓 ∧ (dom 𝑓 ∈ On ∧ ∀𝑦 ∈ dom 𝑓(𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))))
260258, 259bitri 276 . . . . . 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(𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))))
26121, 37, 2603bitri 298 . . . . 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(𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))))
26219, 20, 2613bitr4ri 305 . . . 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 ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦))))))
263262abbi2i 2958 . . 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 ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))}
264263unieqi 4846 . 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 ∧ ∀𝑦𝑥 (𝑓𝑦) = if(𝑦 = ∅, if(𝐴 ∈ V, 𝐴, ∅), if(Lim 𝑦, (𝑓𝑦), (𝐹‘(𝑓 𝑦)))))}
2651, 264eqtr4i 2852 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 207   ∧ wa 396   ∨ wo 843   ∨ w3o 1080   ∧ w3a 1081  ∀wal 1528   = wceq 1530  ∃wex 1773   ∈ wcel 2107  {cab 2804   ≠ wne 3021  ∀wral 3143  ∃wrex 3144  Vcvv 3500   ∖ cdif 3937   ∪ cun 3938   ∩ cin 3939  ∅c0 4295  ifcif 4470  {csn 4564  ⟨cop 4570  ∪ cuni 4837   class class class wbr 5063   I cid 5458   E cep 5463   × cxp 5552  ◡ccnv 5553  dom cdm 5554  ran crn 5555   ↾ cres 5556   “ cima 5557   ∘ ccom 5558  Oncon0 6189  Lim wlim 6190  suc csuc 6191  Fun wfun 6346   Fn wfn 6347  ‘cfv 6352  reccrdg 8036  pprodcpprod 33179   Bigcup cbigcup 33182   Fix cfix 33183   Limits climits 33184   Funs cfuns 33185  Imgcimg 33190  Domaincdomain 33191  Applycapply 33193  Succcsuccf 33196  FullFuncfullfn 33198 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7451 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-pss 3958  df-symdif 4223  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-tp 4569  df-op 4571  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-tr 5170  df-id 5459  df-eprel 5464  df-po 5473  df-so 5474  df-fr 5513  df-we 5515  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-pred 6146  df-ord 6192  df-on 6193  df-lim 6194  df-suc 6195  df-iota 6312  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-fv 6360  df-om 7569  df-1st 7680  df-2nd 7681  df-wrecs 7938  df-recs 7999  df-rdg 8037  df-txp 33202  df-pprod 33203  df-bigcup 33206  df-fix 33207  df-limits 33208  df-funs 33209  df-singleton 33210  df-singles 33211  df-image 33212  df-cart 33213  df-img 33214  df-domain 33215  df-cup 33217  df-succf 33220  df-apply 33221  df-funpart 33222  df-fullfun 33223 This theorem is referenced by: (None)
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