Theorem wfr3g 8321

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 3124	. . . . . . 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 6882	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑤 → (𝐹‘𝑧) = (𝐹‘𝑤))
3		fveq2 6882	. . . . . . . . . . . 12 ⊢ (𝑧 = 𝑤 → (𝐺‘𝑧) = (𝐺‘𝑤))
4	2, 3	eqeq12d 2778	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑤 → ((𝐹‘𝑧) = (𝐺‘𝑧) ↔ (𝐹‘𝑤) = (𝐺‘𝑤)))
5	4	imbi2d 343	. . . . . . . . . 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 3835	. . . . . . . . . . 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 6882	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑧 → (𝐹‘𝑦) = (𝐹‘𝑧))
8		predeq3 6307	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑧 → Pred(𝑅, 𝐴, 𝑦) = Pred(𝑅, 𝐴, 𝑧))
9	8	reseq2d 5976	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑧 → (𝐹 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))
10	9	fveq2d 6886	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑧 → (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))))
11	7, 10	eqeq12d 2778	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑧 → ((𝐹‘𝑦) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝐹‘𝑧) = (𝐻‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))))
12		fveq2 6882	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑧 → (𝐺‘𝑦) = (𝐺‘𝑧))
13	8	reseq2d 5976	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 = 𝑧 → (𝐺 ↾ Pred(𝑅, 𝐴, 𝑦)) = (𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)))
14	13	fveq2d 6886	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 = 𝑧 → (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))))
15	12, 14	eqeq12d 2778	. . . . . . . . . . . . . . . . 17 ⊢ (𝑦 = 𝑧 → ((𝐺‘𝑦) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑦))) ↔ (𝐺‘𝑧) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)))))
16	11, 15	anbi12d 644	. . . . . . . . . . . . . . . 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 3577	. . . . . . . . . . . . . . 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 2784	. . . . . . . . . . . . . . . . . . . 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 464	. . . . . . . . . . . . . . . . . . 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 2784	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝐹‘𝑧) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧))) ∧ (𝐺‘𝑧) = (𝐻‘(𝐺 ↾ Pred(𝑅, 𝐴, 𝑧)))) → (𝐹‘𝑧) = (𝐺‘𝑧))
21	20	ex 418	. . . . . . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . . . . . 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 459	. . . . . . . . . . . . . . . . 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 6311	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Pred(𝑅, 𝐴, 𝑧) ⊆ 𝐴
25		fvreseq 7036	. . . . . . . . . . . . . . . . . . . . 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 704	. . . . . . . . . . . . . . . . . . . 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 483	. . . . . . . . . . . . . . . . . . 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 2768	. . . . . . . . . . . . . . . . . 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 6886	. . . . . . . . . . . . . . . . 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 34	. . . . . . . . . . . . . . . 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 421	. . . . . . . . . . . . . . 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 18	. . . . . . . . . . . . . 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 418	. . . . . . . . . . . . 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 417	. . . . . . . . . . . 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 30	. . . . . . . . . . 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 35	. . . . . . . . . 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 6357	. . . . . . . . 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 3189	. . . . . . . . 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 221	. . . . . . . 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 33	. . . . . . 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 607	. . . . . 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 673	. . . . 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 33	. . . 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 1134	. . 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 2762	. . 3 ⊢ 𝐴 = 𝐴
46	44, 45	jctil 529	. 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 7027	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐺 Fn 𝐴) → (𝐹 = 𝐺 ↔ (𝐴 = 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) = (𝐺‘𝑧))))
48	47	ad2ant2r 760	. . 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 1148	. 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 260	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)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3078   ⊆ wss 3902   Se wse 5610   We wwe 5611   ↾ cres 5661  Predcpred 6302   Fn wfn 6532  ‘cfv 6537

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-pw 4562  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-po 5567  df-so 5568  df-fr 5612  df-se 5613  df-we 5614  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-pred 6303  df-iota 6493  df-fun 6539  df-fn 6540  df-fv 6545

This theorem is used by:  wfr3  8330