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Theorem frr3g 9796
Description: Functions defined by well-founded recursion are identical up to relation, domain, and characteristic function. General version of frr3 9801. (Contributed by Scott Fenton, 10-Feb-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)
Assertion
Ref Expression
frr3g (((𝑅 Fr 𝐴𝑅 Se 𝐴) ∧ (𝐹 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → 𝐹 = 𝐺)
Distinct variable groups:   𝑦,𝐴   𝑦,𝐹   𝑦,𝐺   𝑦,𝐻   𝑦,𝑅

Proof of Theorem frr3g
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ra4v 3885 . . . . . . . . . . 11 (∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (∀𝑦𝐴 (𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹𝑤) = (𝐺𝑤)) → (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (∀𝑦𝐴 (𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → ∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹𝑤) = (𝐺𝑤)))
2 r19.26 3111 . . . . . . . . . . . . . 14 (∀𝑦𝐴 ((𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦)))) ↔ (∀𝑦𝐴 (𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦)))))
32anbi2i 623 . . . . . . . . . . . . 13 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ ∀𝑦𝐴 ((𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) ↔ ((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (∀𝑦𝐴 (𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))))
4 fveq2 6906 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑧 → (𝐹𝑦) = (𝐹𝑧))
5 id 22 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = 𝑧𝑦 = 𝑧)
6 predeq3 6325 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = 𝑧 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑧))
76reseq2d 5997 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = 𝑧 → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))
85, 7oveq12d 7449 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑧 → (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝑧𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))))
94, 8eqeq12d 2753 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑧 → ((𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝐹𝑧) = (𝑧𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))))
10 fveq2 6906 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑧 → (𝐺𝑦) = (𝐺𝑧))
116reseq2d 5997 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = 𝑧 → (𝐺 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)))
125, 11oveq12d 7449 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑧 → (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))))
1310, 12eqeq12d 2753 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑧 → ((𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝐺𝑧) = (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)))))
149, 13anbi12d 632 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑧 → (((𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦)))) ↔ ((𝐹𝑧) = (𝑧𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∧ (𝐺𝑧) = (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))))))
1514rspcva 3620 . . . . . . . . . . . . . . . . 17 ((𝑧𝐴 ∧ ∀𝑦𝐴 ((𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → ((𝐹𝑧) = (𝑧𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∧ (𝐺𝑧) = (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)))))
16 eqtr3 2763 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))) = (𝑧𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∧ (𝐹𝑧) = (𝑧𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))) → (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))) = (𝐹𝑧))
1716eqcomd 2743 . . . . . . . . . . . . . . . . . . . . 21 (((𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))) = (𝑧𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∧ (𝐹𝑧) = (𝑧𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))) → (𝐹𝑧) = (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))))
18 eqtr3 2763 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐹𝑧) = (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))) ∧ (𝐺𝑧) = (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)))) → (𝐹𝑧) = (𝐺𝑧))
1918ex 412 . . . . . . . . . . . . . . . . . . . . 21 ((𝐹𝑧) = (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))) → ((𝐺𝑧) = (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))) → (𝐹𝑧) = (𝐺𝑧)))
2017, 19syl 17 . . . . . . . . . . . . . . . . . . . 20 (((𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))) = (𝑧𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∧ (𝐹𝑧) = (𝑧𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))) → ((𝐺𝑧) = (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))) → (𝐹𝑧) = (𝐺𝑧)))
2120expimpd 453 . . . . . . . . . . . . . . . . . . 19 ((𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))) = (𝑧𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) → (((𝐹𝑧) = (𝑧𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∧ (𝐺𝑧) = (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)))) → (𝐹𝑧) = (𝐺𝑧)))
22 predss 6329 . . . . . . . . . . . . . . . . . . . . . . 23 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝐴
23 fvreseq 7060 . . . . . . . . . . . . . . . . . . . . . . 23 (((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)))
2422, 23mpan2 691 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → ((𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)) = (𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)) ↔ ∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹𝑤) = (𝐺𝑤)))
2524biimpar 477 . . . . . . . . . . . . . . . . . . . . 21 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ ∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹𝑤) = (𝐺𝑤)) → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)) = (𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)))
2625oveq2d 7447 . . . . . . . . . . . . . . . . . . . 20 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ ∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹𝑤) = (𝐺𝑤)) → (𝑧𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) = (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))))
2726eqcomd 2743 . . . . . . . . . . . . . . . . . . 19 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ ∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹𝑤) = (𝐺𝑤)) → (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))) = (𝑧𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))))
2821, 27syl11 33 . . . . . . . . . . . . . . . . . 18 (((𝐹𝑧) = (𝑧𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∧ (𝐺𝑧) = (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)))) → (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ ∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹𝑤) = (𝐺𝑤)) → (𝐹𝑧) = (𝐺𝑧)))
2928expd 415 . . . . . . . . . . . . . . . . 17 (((𝐹𝑧) = (𝑧𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∧ (𝐺𝑧) = (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)))) → ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹𝑤) = (𝐺𝑤) → (𝐹𝑧) = (𝐺𝑧))))
3015, 29syl 17 . . . . . . . . . . . . . . . 16 ((𝑧𝐴 ∧ ∀𝑦𝐴 ((𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹𝑤) = (𝐺𝑤) → (𝐹𝑧) = (𝐺𝑧))))
3130ex 412 . . . . . . . . . . . . . . 15 (𝑧𝐴 → (∀𝑦𝐴 ((𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦)))) → ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹𝑤) = (𝐺𝑤) → (𝐹𝑧) = (𝐺𝑧)))))
3231com23 86 . . . . . . . . . . . . . 14 (𝑧𝐴 → ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (∀𝑦𝐴 ((𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦)))) → (∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹𝑤) = (𝐺𝑤) → (𝐹𝑧) = (𝐺𝑧)))))
3332impd 410 . . . . . . . . . . . . 13 (𝑧𝐴 → (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ ∀𝑦𝐴 ((𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹𝑤) = (𝐺𝑤) → (𝐹𝑧) = (𝐺𝑧))))
343, 33biimtrrid 243 . . . . . . . . . . . 12 (𝑧𝐴 → (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (∀𝑦𝐴 (𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹𝑤) = (𝐺𝑤) → (𝐹𝑧) = (𝐺𝑧))))
3534a2d 29 . . . . . . . . . . 11 (𝑧𝐴 → ((((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (∀𝑦𝐴 (𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → ∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹𝑤) = (𝐺𝑤)) → (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (∀𝑦𝐴 (𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹𝑧) = (𝐺𝑧))))
361, 35syl5 34 . . . . . . . . . 10 (𝑧𝐴 → (∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (∀𝑦𝐴 (𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹𝑤) = (𝐺𝑤)) → (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (∀𝑦𝐴 (𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹𝑧) = (𝐺𝑧))))
37 fveq2 6906 . . . . . . . . . . . 12 (𝑧 = 𝑤 → (𝐹𝑧) = (𝐹𝑤))
38 fveq2 6906 . . . . . . . . . . . 12 (𝑧 = 𝑤 → (𝐺𝑧) = (𝐺𝑤))
3937, 38eqeq12d 2753 . . . . . . . . . . 11 (𝑧 = 𝑤 → ((𝐹𝑧) = (𝐺𝑧) ↔ (𝐹𝑤) = (𝐺𝑤)))
4039imbi2d 340 . . . . . . . . . 10 (𝑧 = 𝑤 → ((((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (∀𝑦𝐴 (𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹𝑧) = (𝐺𝑧)) ↔ (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (∀𝑦𝐴 (𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹𝑤) = (𝐺𝑤))))
4136, 40frins2 9794 . . . . . . . . 9 ((𝑅 Fr 𝐴𝑅 Se 𝐴) → ∀𝑧𝐴 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (∀𝑦𝐴 (𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹𝑧) = (𝐺𝑧)))
42 rsp 3247 . . . . . . . . 9 (∀𝑧𝐴 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (∀𝑦𝐴 (𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹𝑧) = (𝐺𝑧)) → (𝑧𝐴 → (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (∀𝑦𝐴 (𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹𝑧) = (𝐺𝑧))))
4341, 42syl 17 . . . . . . . 8 ((𝑅 Fr 𝐴𝑅 Se 𝐴) → (𝑧𝐴 → (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (∀𝑦𝐴 (𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹𝑧) = (𝐺𝑧))))
4443com3r 87 . . . . . . 7 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ (∀𝑦𝐴 (𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → ((𝑅 Fr 𝐴𝑅 Se 𝐴) → (𝑧𝐴 → (𝐹𝑧) = (𝐺𝑧))))
4544an4s 660 . . . . . 6 (((𝐹 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → ((𝑅 Fr 𝐴𝑅 Se 𝐴) → (𝑧𝐴 → (𝐹𝑧) = (𝐺𝑧))))
4645com12 32 . . . . 5 ((𝑅 Fr 𝐴𝑅 Se 𝐴) → (((𝐹 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝑧𝐴 → (𝐹𝑧) = (𝐺𝑧))))
47463impib 1117 . . . 4 (((𝑅 Fr 𝐴𝑅 Se 𝐴) ∧ (𝐹 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝑧𝐴 → (𝐹𝑧) = (𝐺𝑧)))
4847ralrimiv 3145 . . 3 (((𝑅 Fr 𝐴𝑅 Se 𝐴) ∧ (𝐹 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → ∀𝑧𝐴 (𝐹𝑧) = (𝐺𝑧))
49 eqid 2737 . . 3 𝐴 = 𝐴
5048, 49jctil 519 . 2 (((𝑅 Fr 𝐴𝑅 Se 𝐴) ∧ (𝐹 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐴 = 𝐴 ∧ ∀𝑧𝐴 (𝐹𝑧) = (𝐺𝑧)))
51 eqfnfv2 7052 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ (𝐴 = 𝐴 ∧ ∀𝑧𝐴 (𝐹𝑧) = (𝐺𝑧))))
5251ad2ant2r 747 . . 3 (((𝐹 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹 = 𝐺 ↔ (𝐴 = 𝐴 ∧ ∀𝑧𝐴 (𝐹𝑧) = (𝐺𝑧))))
53523adant1 1131 . 2 (((𝑅 Fr 𝐴𝑅 Se 𝐴) ∧ (𝐹 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹 = 𝐺 ↔ (𝐴 = 𝐴 ∧ ∀𝑧𝐴 (𝐹𝑧) = (𝐺𝑧))))
5450, 53mpbird 257 1 (((𝑅 Fr 𝐴𝑅 Se 𝐴) ∧ (𝐹 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦𝐴 (𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → 𝐹 = 𝐺)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wss 3951   Fr wfr 5634   Se wse 5635  cres 5687  Predcpred 6320   Fn wfn 6556  cfv 6561  (class class class)co 7431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755  ax-inf2 9681
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-pss 3971  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-tr 5260  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5637  df-se 5638  df-we 5639  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-pred 6321  df-ord 6387  df-on 6388  df-lim 6389  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-om 7888  df-2nd 8015  df-frecs 8306  df-wrecs 8337  df-recs 8411  df-rdg 8450  df-1o 8506  df-oadd 8510  df-ttrcl 9748
This theorem is referenced by:  frrlem15  9797  frr3  9801
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