Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  frr3g Structured version   Visualization version   GIF version

Theorem frr3g 30858
Description: Functions defined by founded recursion are identical up to relation, domain, and characteristic function. General version of frr3. (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 3394 . . . . . . . . . . 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 2950 . . . . . . . . . . . . . 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 725 . . . . . . . . . . . . 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 5987 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑧 → (𝐹𝑦) = (𝐹𝑧))
5 id 22 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = 𝑧𝑦 = 𝑧)
6 predeq3 5491 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = 𝑧 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑧))
76reseq2d 5208 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = 𝑧 → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))
85, 7oveq12d 6444 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑧 → (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝑧𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))))
94, 8eqeq12d 2529 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑧 → ((𝐹𝑦) = (𝑦𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝐹𝑧) = (𝑧𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))))
10 fveq2 5987 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑧 → (𝐺𝑦) = (𝐺𝑧))
116reseq2d 5208 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = 𝑧 → (𝐺 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)))
125, 11oveq12d 6444 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝑧 → (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))))
1310, 12eqeq12d 2529 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑧 → ((𝐺𝑦) = (𝑦𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝐺𝑧) = (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)))))
149, 13anbi12d 742 . . . . . . . . . . . . . . . . . 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 3184 . . . . . . . . . . . . . . . . 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 predss 5494 . . . . . . . . . . . . . . . . . . . . . . . 24 Pred(𝑅, 𝐴, 𝑧) ⊆ 𝐴
17 fvreseq 6111 . . . . . . . . . . . . . . . . . . . . . . . 24 (((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)))
1816, 17mpan2 702 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → ((𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)) = (𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)) ↔ ∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹𝑤) = (𝐺𝑤)))
1918biimpar 500 . . . . . . . . . . . . . . . . . . . . . 22 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ ∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹𝑤) = (𝐺𝑤)) → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)) = (𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)))
2019oveq2d 6442 . . . . . . . . . . . . . . . . . . . . 21 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ ∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹𝑤) = (𝐺𝑤)) → (𝑧𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) = (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))))
2120eqcomd 2520 . . . . . . . . . . . . . . . . . . . 20 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ ∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹𝑤) = (𝐺𝑤)) → (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))) = (𝑧𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))))
22 eqtr3 2535 . . . . . . . . . . . . . . . . . . . . . . 23 (((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))
2322eqcomd 2520 . . . . . . . . . . . . . . . . . . . . . 22 (((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))))
24 eqtr3 2535 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝐹𝑧) = (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))) ∧ (𝐺𝑧) = (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)))) → (𝐹𝑧) = (𝐺𝑧))
2524ex 448 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐹𝑧) = (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))) → ((𝐺𝑧) = (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))) → (𝐹𝑧) = (𝐺𝑧)))
2623, 25syl 17 . . . . . . . . . . . . . . . . . . . . 21 (((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)))
2726expimpd 626 . . . . . . . . . . . . . . . . . . . 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)))
2821, 27syl 17 . . . . . . . . . . . . . . . . . . 19 (((𝐹 Fn 𝐴𝐺 Fn 𝐴) ∧ ∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹𝑤) = (𝐺𝑤)) → (((𝐹𝑧) = (𝑧𝐻(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∧ (𝐺𝑧) = (𝑧𝐻(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)))) → (𝐹𝑧) = (𝐺𝑧)))
2928com12 32 . . . . . . . . . . . . . . . . . 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)))
3029expd 450 . . . . . . . . . . . . . . . . 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))))
3115, 30syl 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))))
3231ex 448 . . . . . . . . . . . . . . 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)))))
3332com23 83 . . . . . . . . . . . . . 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)))))
3433impd 445 . . . . . . . . . . . . 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))))
353, 34syl5bir 231 . . . . . . . . . . . 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))))
3635a2d 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))))
371, 36syl5 33 . . . . . . . . . 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))))
38 fveq2 5987 . . . . . . . . . . . 12 (𝑧 = 𝑤 → (𝐹𝑧) = (𝐹𝑤))
39 fveq2 5987 . . . . . . . . . . . 12 (𝑧 = 𝑤 → (𝐺𝑧) = (𝐺𝑤))
4038, 39eqeq12d 2529 . . . . . . . . . . 11 (𝑧 = 𝑤 → ((𝐹𝑧) = (𝐺𝑧) ↔ (𝐹𝑤) = (𝐺𝑤)))
4140imbi2d 328 . . . . . . . . . 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))))
4237, 41frins2g 30832 . . . . . . . . 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)))
43 rsp 2817 . . . . . . . . 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))))
4442, 43syl 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))))
4544com3r 84 . . . . . . 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))))
4645an4s 864 . . . . . 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))))
4746com12 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))))
48473impib 1253 . . . 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)))
4948ralrimiv 2852 . . 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))
50 eqid 2514 . . 3 𝐴 = 𝐴
5149, 50jctil 557 . 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)))
52 eqfnfv2 6104 . . . 4 ((𝐹 Fn 𝐴𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ (𝐴 = 𝐴 ∧ ∀𝑧𝐴 (𝐹𝑧) = (𝐺𝑧))))
5352ad2ant2r 778 . . 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))))
54533adant1 1071 . 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))))
5551, 54mpbird 245 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 194  wa 382  w3a 1030   = wceq 1474  wcel 1938  wral 2800  wss 3444   Fr wfr 4888   Se wse 4889  cres 4934  Predcpred 5486   Fn wfn 5684  cfv 5689  (class class class)co 6426
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-rep 4597  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732  ax-un 6723  ax-inf2 8297
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-ral 2805  df-rex 2806  df-reu 2807  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-pss 3460  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-tp 4033  df-op 4035  df-uni 4271  df-iun 4355  df-br 4482  df-opab 4542  df-mpt 4543  df-tr 4579  df-eprel 4843  df-id 4847  df-po 4853  df-so 4854  df-fr 4891  df-se 4892  df-we 4893  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-pred 5487  df-ord 5533  df-on 5534  df-lim 5535  df-suc 5536  df-iota 5653  df-fun 5691  df-fn 5692  df-f 5693  df-f1 5694  df-fo 5695  df-f1o 5696  df-fv 5697  df-ov 6429  df-om 6834  df-wrecs 7169  df-recs 7231  df-rdg 7269  df-trpred 30804
This theorem is referenced by:  frrlem5  30863
  Copyright terms: Public domain W3C validator