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Theorem wfr3g 7458
Description: Functions defined by well-founded recursion are identical up to relation, domain, and characteristic function. (Contributed by Scott Fenton, 11-Feb-2011.)
Assertion
Ref Expression
wfr3g (((𝑅 We 𝐴𝑅 Se 𝐴) ∧ (𝐹 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐹𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → 𝐹 = 𝐺)
Distinct variable groups:   𝑦,𝐴   𝑦,𝐹   𝑦,𝐺   𝑦,𝐻   𝑦,𝑅

Proof of Theorem wfr3g
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 r19.26 3093 . . . . . . 7 (∀𝑦𝐴 ((𝐹𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦)))) ↔ (∀𝑦𝐴 (𝐹𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦)))))
2 fveq2 6229 . . . . . . . . . . . 12 (𝑧 = 𝑤 → (𝐹𝑧) = (𝐹𝑤))
3 fveq2 6229 . . . . . . . . . . . 12 (𝑧 = 𝑤 → (𝐺𝑧) = (𝐺𝑤))
42, 3eqeq12d 2666 . . . . . . . . . . 11 (𝑧 = 𝑤 → ((𝐹𝑧) = (𝐺𝑧) ↔ (𝐹𝑤) = (𝐺𝑤)))
54imbi2d 329 . . . . . . . . . 10 (𝑧 = 𝑤 → ((((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ ∀𝑦𝐴 ((𝐹𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹𝑧) = (𝐺𝑧)) ↔ (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ ∀𝑦𝐴 ((𝐹𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹𝑤) = (𝐺𝑤))))
6 ra4v 3557 . . . . . . . . . . 11 (∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ ∀𝑦𝐴 ((𝐹𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹𝑤) = (𝐺𝑤)) → (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ ∀𝑦𝐴 ((𝐹𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → ∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹𝑤) = (𝐺𝑤)))
7 fveq2 6229 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑧 → (𝐹𝑦) = (𝐹𝑧))
8 predeq3 5722 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = 𝑧 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑧))
98reseq2d 5428 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑧 → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))
109fveq2d 6233 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑧 → (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))))
117, 10eqeq12d 2666 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑧 → ((𝐹𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝐹𝑧) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))))
12 fveq2 6229 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑧 → (𝐺𝑦) = (𝐺𝑧))
138reseq2d 5428 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑧 → (𝐺 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)))
1413fveq2d 6233 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑧 → (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))))
1512, 14eqeq12d 2666 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑧 → ((𝐺𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝐺𝑧) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)))))
1611, 15anbi12d 747 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑧 → (((𝐹𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦)))) ↔ ((𝐹𝑧) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∧ (𝐺𝑧) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))))))
1716rspcva 3338 . . . . . . . . . . . . . . . 16 ((𝑧𝐴 ∧ ∀𝑦𝐴 ((𝐹𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → ((𝐹𝑧) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∧ (𝐺𝑧) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)))))
18 eqtr3 2672 . . . . . . . . . . . . . . . . . . . . 21 (((𝐹𝑧) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∧ (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))) → (𝐹𝑧) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))))
1918ancoms 468 . . . . . . . . . . . . . . . . . . . 20 (((𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∧ (𝐹𝑧) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))) → (𝐹𝑧) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))))
20 eqtr3 2672 . . . . . . . . . . . . . . . . . . . . 21 (((𝐹𝑧) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))) ∧ (𝐺𝑧) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)))) → (𝐹𝑧) = (𝐺𝑧))
2120ex 449 . . . . . . . . . . . . . . . . . . . 20 ((𝐹𝑧) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))) → ((𝐺𝑧) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))) → (𝐹𝑧) = (𝐺𝑧)))
2219, 21syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∧ (𝐹𝑧) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))) → ((𝐺𝑧) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))) → (𝐹𝑧) = (𝐺𝑧)))
2322expimpd 628 . . . . . . . . . . . . . . . . . 18 ((𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) → (((𝐹𝑧) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∧ (𝐺𝑧) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)))) → (𝐹𝑧) = (𝐺𝑧)))
24 predss 5725 . . . . . . . . . . . . . . . . . . . . . 22 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝐴
25 fvreseq 6359 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ Pred(𝑅, 𝐴, 𝑧) ⊆ 𝐴) → ((𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)) = (𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)) ↔ ∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹𝑤) = (𝐺𝑤)))
2624, 25mpan2 707 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → ((𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)) = (𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)) ↔ ∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹𝑤) = (𝐺𝑤)))
2726biimpar 501 . . . . . . . . . . . . . . . . . . . 20 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ ∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹𝑤) = (𝐺𝑤)) → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)) = (𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)))
2827eqcomd 2657 . . . . . . . . . . . . . . . . . . 19 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ ∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹𝑤) = (𝐺𝑤)) → (𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)) = (𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))
2928fveq2d 6233 . . . . . . . . . . . . . . . . . 18 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ ∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹𝑤) = (𝐺𝑤)) → (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))))
3023, 29syl11 33 . . . . . . . . . . . . . . . . 17 (((𝐹𝑧) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∧ (𝐺𝑧) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)))) → (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ ∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹𝑤) = (𝐺𝑤)) → (𝐹𝑧) = (𝐺𝑧)))
3130expd 451 . . . . . . . . . . . . . . . 16 (((𝐹𝑧) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∧ (𝐺𝑧) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)))) → ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹𝑤) = (𝐺𝑤) → (𝐹𝑧) = (𝐺𝑧))))
3217, 31syl 17 . . . . . . . . . . . . . . 15 ((𝑧𝐴 ∧ ∀𝑦𝐴 ((𝐹𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹𝑤) = (𝐺𝑤) → (𝐹𝑧) = (𝐺𝑧))))
3332ex 449 . . . . . . . . . . . . . 14 (𝑧𝐴 → (∀𝑦𝐴 ((𝐹𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦)))) → ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹𝑤) = (𝐺𝑤) → (𝐹𝑧) = (𝐺𝑧)))))
3433com23 86 . . . . . . . . . . . . 13 (𝑧𝐴 → ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (∀𝑦𝐴 ((𝐹𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦)))) → (∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹𝑤) = (𝐺𝑤) → (𝐹𝑧) = (𝐺𝑧)))))
3534impd 446 . . . . . . . . . . . 12 (𝑧𝐴 → (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ ∀𝑦𝐴 ((𝐹𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹𝑤) = (𝐺𝑤) → (𝐹𝑧) = (𝐺𝑧))))
3635a2d 29 . . . . . . . . . . 11 (𝑧𝐴 → ((((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ ∀𝑦𝐴 ((𝐹𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → ∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹𝑤) = (𝐺𝑤)) → (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ ∀𝑦𝐴 ((𝐹𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹𝑧) = (𝐺𝑧))))
376, 36syl5 34 . . . . . . . . . 10 (𝑧𝐴 → (∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ ∀𝑦𝐴 ((𝐹𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹𝑤) = (𝐺𝑤)) → (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ ∀𝑦𝐴 ((𝐹𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹𝑧) = (𝐺𝑧))))
385, 37wfis2g 5757 . . . . . . . . 9 ((𝑅 We 𝐴𝑅 Se 𝐴) → ∀𝑧𝐴 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ ∀𝑦𝐴 ((𝐹𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹𝑧) = (𝐺𝑧)))
39 r19.21v 2989 . . . . . . . . 9 (∀𝑧𝐴 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ ∀𝑦𝐴 ((𝐹𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹𝑧) = (𝐺𝑧)) ↔ (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ ∀𝑦𝐴 ((𝐹𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → ∀𝑧𝐴 (𝐹𝑧) = (𝐺𝑧)))
4038, 39sylib 208 . . . . . . . 8 ((𝑅 We 𝐴𝑅 Se 𝐴) → (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ ∀𝑦𝐴 ((𝐹𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → ∀𝑧𝐴 (𝐹𝑧) = (𝐺𝑧)))
4140com12 32 . . . . . . 7 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ ∀𝑦𝐴 ((𝐹𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → ((𝑅 We 𝐴𝑅 Se 𝐴) → ∀𝑧𝐴 (𝐹𝑧) = (𝐺𝑧)))
421, 41sylan2br 492 . . . . . 6 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (∀𝑦𝐴 (𝐹𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → ((𝑅 We 𝐴𝑅 Se 𝐴) → ∀𝑧𝐴 (𝐹𝑧) = (𝐺𝑧)))
4342an4s 886 . . . . 5 (((𝐹 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐹𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → ((𝑅 We 𝐴𝑅 Se 𝐴) → ∀𝑧𝐴 (𝐹𝑧) = (𝐺𝑧)))
4443com12 32 . . . 4 ((𝑅 We 𝐴𝑅 Se 𝐴) → (((𝐹 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐹𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → ∀𝑧𝐴 (𝐹𝑧) = (𝐺𝑧)))
45443impib 1281 . . 3 (((𝑅 We 𝐴𝑅 Se 𝐴) ∧ (𝐹 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐹𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → ∀𝑧𝐴 (𝐹𝑧) = (𝐺𝑧))
46 eqid 2651 . . 3 𝐴 = 𝐴
4745, 46jctil 559 . 2 (((𝑅 We 𝐴𝑅 Se 𝐴) ∧ (𝐹 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐹𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐴 = 𝐴 ∧ ∀𝑧𝐴 (𝐹𝑧) = (𝐺𝑧)))
48 eqfnfv2 6352 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ (𝐴 = 𝐴 ∧ ∀𝑧𝐴 (𝐹𝑧) = (𝐺𝑧))))
4948ad2ant2r 798 . . 3 (((𝐹 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐹𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹 = 𝐺 ↔ (𝐴 = 𝐴 ∧ ∀𝑧𝐴 (𝐹𝑧) = (𝐺𝑧))))
50493adant1 1099 . 2 (((𝑅 We 𝐴𝑅 Se 𝐴) ∧ (𝐹 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐹𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹 = 𝐺 ↔ (𝐴 = 𝐴 ∧ ∀𝑧𝐴 (𝐹𝑧) = (𝐺𝑧))))
5147, 50mpbird 247 1 (((𝑅 We 𝐴𝑅 Se 𝐴) ∧ (𝐹 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐹𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → 𝐹 = 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  wss 3607   Se wse 5100   We wwe 5101  cres 5145  Predcpred 5717   Fn wfn 5921  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rmo 2949  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-po 5064  df-so 5065  df-fr 5102  df-se 5103  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934
This theorem is referenced by:  wfrlem5  7464  wfr3  7480
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