Theorem wfr3g 8328

Description: Functions defined by well-ordered recursion are identical up to relation, domain, and characteristic function. (Contributed by Scott Fenton, 11-Feb-2011.)

Assertion

Ref	Expression
wfr3g	⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐹 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (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.

Step	Hyp	Ref	Expression
1		r19.26 3108	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝐴 ((𝐹‘𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺‘𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦)))) ↔ (∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦)))))
2		fveq2 6897	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑤 → (𝐹‘𝑧) = (𝐹‘𝑤))
3		fveq2 6897	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑤 → (𝐺‘𝑧) = (𝐺‘𝑤))
4	2, 3	eqeq12d 2744	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → ((𝐹‘𝑧) = (𝐺‘𝑧) ↔ (𝐹‘𝑤) = (𝐺‘𝑤)))
5	4	imbi2d 340	. . . . . . . . . 10 ⊢ (𝑧 = 𝑤 → ((((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ ∀𝑦 ∈ 𝐴 ((𝐹‘𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺‘𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹‘𝑧) = (𝐺‘𝑧)) ↔ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ ∀𝑦 ∈ 𝐴 ((𝐹‘𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺‘𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹‘𝑤) = (𝐺‘𝑤))))
6		ra4v 3878	. . . . . . . . . . 11 ⊢ (∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ ∀𝑦 ∈ 𝐴 ((𝐹‘𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺‘𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹‘𝑤) = (𝐺‘𝑤)) → (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ ∀𝑦 ∈ 𝐴 ((𝐹‘𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺‘𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → ∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹‘𝑤) = (𝐺‘𝑤)))
7		fveq2 6897	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑧 → (𝐹‘𝑦) = (𝐹‘𝑧))
8		predeq3 6309	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑧 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑧))
9	8	reseq2d 5985	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑧 → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))
10	9	fveq2d 6901	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑧 → (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))))
11	7, 10	eqeq12d 2744	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑧 → ((𝐹‘𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝐹‘𝑧) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))))
12		fveq2 6897	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑧 → (𝐺‘𝑦) = (𝐺‘𝑧))
13	8	reseq2d 5985	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑧 → (𝐺 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)))
14	13	fveq2d 6901	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑧 → (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))))
15	12, 14	eqeq12d 2744	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑧 → ((𝐺‘𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝐺‘𝑧) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)))))
16	11, 15	anbi12d 631	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 = 𝑧 → (((𝐹‘𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺‘𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦)))) ↔ ((𝐹‘𝑧) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∧ (𝐺‘𝑧) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))))))
17	16	rspcva 3607	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ((𝐹‘𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺‘𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → ((𝐹‘𝑧) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∧ (𝐺‘𝑧) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)))))
18		eqtr3 2754	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐹‘𝑧) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∧ (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))) → (𝐹‘𝑧) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))))
19	18	ancoms 458	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∧ (𝐹‘𝑧) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))) → (𝐹‘𝑧) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))))
20		eqtr3 2754	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐹‘𝑧) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))) ∧ (𝐺‘𝑧) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)))) → (𝐹‘𝑧) = (𝐺‘𝑧))
21	20	ex 412	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹‘𝑧) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))) → ((𝐺‘𝑧) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))) → (𝐹‘𝑧) = (𝐺‘𝑧)))
22	19, 21	syl 17	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∧ (𝐹‘𝑧) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))) → ((𝐺‘𝑧) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))) → (𝐹‘𝑧) = (𝐺‘𝑧)))
23	22	expimpd 453	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) → (((𝐹‘𝑧) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∧ (𝐺‘𝑧) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)))) → (𝐹‘𝑧) = (𝐺‘𝑧)))
24		predss 6313	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Pred(𝑅, 𝐴, 𝑧) ⊆ 𝐴
25		fvreseq 7049	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ Pred(𝑅, 𝐴, 𝑧) ⊆ 𝐴) → ((𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)) = (𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)) ↔ ∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹‘𝑤) = (𝐺‘𝑤)))
26	24, 25	mpan2 690	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → ((𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)) = (𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)) ↔ ∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹‘𝑤) = (𝐺‘𝑤)))
27	26	biimpar 477	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ ∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹‘𝑤) = (𝐺‘𝑤)) → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)) = (𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)))
28	27	eqcomd 2734	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ ∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹‘𝑤) = (𝐺‘𝑤)) → (𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)) = (𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))
29	28	fveq2d 6901	. . . . . . . . . . . . . . . . 17 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ ∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹‘𝑤) = (𝐺‘𝑤)) → (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))))
30	23, 29	syl11 33	. . . . . . . . . . . . . . . 16 ⊢ (((𝐹‘𝑧) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∧ (𝐺‘𝑧) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)))) → (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ ∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹‘𝑤) = (𝐺‘𝑤)) → (𝐹‘𝑧) = (𝐺‘𝑧)))
31	30	expd 415	. . . . . . . . . . . . . . 15 ⊢ (((𝐹‘𝑧) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∧ (𝐺‘𝑧) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)))) → ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹‘𝑤) = (𝐺‘𝑤) → (𝐹‘𝑧) = (𝐺‘𝑧))))
32	17, 31	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐴 ((𝐹‘𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺‘𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹‘𝑤) = (𝐺‘𝑤) → (𝐹‘𝑧) = (𝐺‘𝑧))))
33	32	ex 412	. . . . . . . . . . . . 13 ⊢ (𝑧 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 ((𝐹‘𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺‘𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦)))) → ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹‘𝑤) = (𝐺‘𝑤) → (𝐹‘𝑧) = (𝐺‘𝑧)))))
34	33	impcomd 411	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝐴 → (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ ∀𝑦 ∈ 𝐴 ((𝐹‘𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺‘𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹‘𝑤) = (𝐺‘𝑤) → (𝐹‘𝑧) = (𝐺‘𝑧))))
35	34	a2d 29	. . . . . . . . . . 11 ⊢ (𝑧 ∈ 𝐴 → ((((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ ∀𝑦 ∈ 𝐴 ((𝐹‘𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺‘𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → ∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(𝐹‘𝑤) = (𝐺‘𝑤)) → (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ ∀𝑦 ∈ 𝐴 ((𝐹‘𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺‘𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹‘𝑧) = (𝐺‘𝑧))))
36	6, 35	syl5 34	. . . . . . . . . 10 ⊢ (𝑧 ∈ 𝐴 → (∀𝑤 ∈ Pred (𝑅, 𝐴, 𝑧)(((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ ∀𝑦 ∈ 𝐴 ((𝐹‘𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺‘𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹‘𝑤) = (𝐺‘𝑤)) → (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ ∀𝑦 ∈ 𝐴 ((𝐹‘𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺‘𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹‘𝑧) = (𝐺‘𝑧))))
37	5, 36	wfis2g 6365	. . . . . . . . 9 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑧 ∈ 𝐴 (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ ∀𝑦 ∈ 𝐴 ((𝐹‘𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺‘𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹‘𝑧) = (𝐺‘𝑧)))
38		r19.21v 3176	. . . . . . . . 9 ⊢ (∀𝑧 ∈ 𝐴 (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ ∀𝑦 ∈ 𝐴 ((𝐹‘𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺‘𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹‘𝑧) = (𝐺‘𝑧)) ↔ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ ∀𝑦 ∈ 𝐴 ((𝐹‘𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺‘𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) = (𝐺‘𝑧)))
39	37, 38	sylib 217	. . . . . . . 8 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ ∀𝑦 ∈ 𝐴 ((𝐹‘𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺‘𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) = (𝐺‘𝑧)))
40	39	com12 32	. . . . . . 7 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ ∀𝑦 ∈ 𝐴 ((𝐹‘𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ (𝐺‘𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) = (𝐺‘𝑧)))
41	1, 40	sylan2br 594	. . . . . 6 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) ∧ (∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) = (𝐺‘𝑧)))
42	41	an4s 659	. . . . 5 ⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) = (𝐺‘𝑧)))
43	42	com12 32	. . . 4 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → (((𝐹 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) = (𝐺‘𝑧)))
44	43	3impib 1114	. . 3 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐹 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) = (𝐺‘𝑧))
45		eqid 2728	. . 3 ⊢ 𝐴 = 𝐴
46	44, 45	jctil 519	. 2 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐹 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐴 = 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) = (𝐺‘𝑧)))
47		eqfnfv2 7041	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ (𝐴 = 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) = (𝐺‘𝑧))))
48	47	ad2ant2r 746	. . 3 ⊢ (((𝐹 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹 = 𝐺 ↔ (𝐴 = 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) = (𝐺‘𝑧))))
49	48	3adant1 1128	. 2 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐹 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → (𝐹 = 𝐺 ↔ (𝐴 = 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) = (𝐺‘𝑧))))
50	46, 49	mpbird 257	1 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐹 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐹‘𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)))) ∧ (𝐺 Fn 𝐴 ∧ ∀𝑦 ∈ 𝐴 (𝐺‘𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))))) → 𝐹 = 𝐺)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099  ∀wral 3058   ⊆ wss 3947   Se wse 5631   We wwe 5632   ↾ cres 5680  Predcpred 6304   Fn wfn 6543  ‘cfv 6548

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-po 5590  df-so 5591  df-fr 5633  df-se 5634  df-we 5635  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-pred 6305  df-iota 6500  df-fun 6550  df-fn 6551  df-fv 6556

This theorem is referenced by:  wfrlem5OLD  8334  wfr3  8358  wfr3OLD  8359