Theorem hashfacenOLD 14410

Description: Obsolete version of hashfacen 14409 as of 7-Aug-2024. (Contributed by Mario Carneiro, 21-Jan-2015.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
hashfacenOLD	⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐶} ≈ {𝑓 ∣ 𝑓:𝐵–1-1-onto→𝐷})

Distinct variable groups:   𝐴,𝑓   𝐵,𝑓   𝐶,𝑓   𝐷,𝑓

Proof of Theorem hashfacenOLD

Dummy variables 𝑔 ℎ 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bren 8944	. 2 ⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑔 𝑔:𝐴–1-1-onto→𝐵)
2		bren 8944	. 2 ⊢ (𝐶 ≈ 𝐷 ↔ ∃ℎ ℎ:𝐶–1-1-onto→𝐷)
3		exdistrv 1951	. . 3 ⊢ (∃𝑔∃ℎ(𝑔:𝐴–1-1-onto→𝐵 ∧ ℎ:𝐶–1-1-onto→𝐷) ↔ (∃𝑔 𝑔:𝐴–1-1-onto→𝐵 ∧ ∃ℎ ℎ:𝐶–1-1-onto→𝐷))
4		f1of 6823	. . . . . . . 8 ⊢ (𝑓:𝐴–1-1-onto→𝐶 → 𝑓:𝐴⟶𝐶)
5		f1odm 6827	. . . . . . . . . 10 ⊢ (ℎ:𝐶–1-1-onto→𝐷 → dom ℎ = 𝐶)
6		vex 3470	. . . . . . . . . . 11 ⊢ ℎ ∈ V
7	6	dmex 7895	. . . . . . . . . 10 ⊢ dom ℎ ∈ V
8	5, 7	eqeltrrdi 2834	. . . . . . . . 9 ⊢ (ℎ:𝐶–1-1-onto→𝐷 → 𝐶 ∈ V)
9		f1odm 6827	. . . . . . . . . 10 ⊢ (𝑔:𝐴–1-1-onto→𝐵 → dom 𝑔 = 𝐴)
10		vex 3470	. . . . . . . . . . 11 ⊢ 𝑔 ∈ V
11	10	dmex 7895	. . . . . . . . . 10 ⊢ dom 𝑔 ∈ V
12	9, 11	eqeltrrdi 2834	. . . . . . . . 9 ⊢ (𝑔:𝐴–1-1-onto→𝐵 → 𝐴 ∈ V)
13		elmapg 8828	. . . . . . . . 9 ⊢ ((𝐶 ∈ V ∧ 𝐴 ∈ V) → (𝑓 ∈ (𝐶 ↑m 𝐴) ↔ 𝑓:𝐴⟶𝐶))
14	8, 12, 13	syl2anr 596	. . . . . . . 8 ⊢ ((𝑔:𝐴–1-1-onto→𝐵 ∧ ℎ:𝐶–1-1-onto→𝐷) → (𝑓 ∈ (𝐶 ↑m 𝐴) ↔ 𝑓:𝐴⟶𝐶))
15	4, 14	imbitrrid 245	. . . . . . 7 ⊢ ((𝑔:𝐴–1-1-onto→𝐵 ∧ ℎ:𝐶–1-1-onto→𝐷) → (𝑓:𝐴–1-1-onto→𝐶 → 𝑓 ∈ (𝐶 ↑m 𝐴)))
16	15	abssdv 4057	. . . . . 6 ⊢ ((𝑔:𝐴–1-1-onto→𝐵 ∧ ℎ:𝐶–1-1-onto→𝐷) → {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐶} ⊆ (𝐶 ↑m 𝐴))
17		ovex 7434	. . . . . . 7 ⊢ (𝐶 ↑m 𝐴) ∈ V
18	17	ssex 5311	. . . . . 6 ⊢ ({𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐶} ⊆ (𝐶 ↑m 𝐴) → {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐶} ∈ V)
19	16, 18	syl 17	. . . . 5 ⊢ ((𝑔:𝐴–1-1-onto→𝐵 ∧ ℎ:𝐶–1-1-onto→𝐷) → {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐶} ∈ V)
20		f1of 6823	. . . . . . . 8 ⊢ (𝑓:𝐵–1-1-onto→𝐷 → 𝑓:𝐵⟶𝐷)
21		f1ofo 6830	. . . . . . . . . . 11 ⊢ (ℎ:𝐶–1-1-onto→𝐷 → ℎ:𝐶–onto→𝐷)
22		forn 6798	. . . . . . . . . . 11 ⊢ (ℎ:𝐶–onto→𝐷 → ran ℎ = 𝐷)
23	21, 22	syl 17	. . . . . . . . . 10 ⊢ (ℎ:𝐶–1-1-onto→𝐷 → ran ℎ = 𝐷)
24	6	rnex 7896	. . . . . . . . . 10 ⊢ ran ℎ ∈ V
25	23, 24	eqeltrrdi 2834	. . . . . . . . 9 ⊢ (ℎ:𝐶–1-1-onto→𝐷 → 𝐷 ∈ V)
26		f1ofo 6830	. . . . . . . . . . 11 ⊢ (𝑔:𝐴–1-1-onto→𝐵 → 𝑔:𝐴–onto→𝐵)
27		forn 6798	. . . . . . . . . . 11 ⊢ (𝑔:𝐴–onto→𝐵 → ran 𝑔 = 𝐵)
28	26, 27	syl 17	. . . . . . . . . 10 ⊢ (𝑔:𝐴–1-1-onto→𝐵 → ran 𝑔 = 𝐵)
29	10	rnex 7896	. . . . . . . . . 10 ⊢ ran 𝑔 ∈ V
30	28, 29	eqeltrrdi 2834	. . . . . . . . 9 ⊢ (𝑔:𝐴–1-1-onto→𝐵 → 𝐵 ∈ V)
31		elmapg 8828	. . . . . . . . 9 ⊢ ((𝐷 ∈ V ∧ 𝐵 ∈ V) → (𝑓 ∈ (𝐷 ↑m 𝐵) ↔ 𝑓:𝐵⟶𝐷))
32	25, 30, 31	syl2anr 596	. . . . . . . 8 ⊢ ((𝑔:𝐴–1-1-onto→𝐵 ∧ ℎ:𝐶–1-1-onto→𝐷) → (𝑓 ∈ (𝐷 ↑m 𝐵) ↔ 𝑓:𝐵⟶𝐷))
33	20, 32	imbitrrid 245	. . . . . . 7 ⊢ ((𝑔:𝐴–1-1-onto→𝐵 ∧ ℎ:𝐶–1-1-onto→𝐷) → (𝑓:𝐵–1-1-onto→𝐷 → 𝑓 ∈ (𝐷 ↑m 𝐵)))
34	33	abssdv 4057	. . . . . 6 ⊢ ((𝑔:𝐴–1-1-onto→𝐵 ∧ ℎ:𝐶–1-1-onto→𝐷) → {𝑓 ∣ 𝑓:𝐵–1-1-onto→𝐷} ⊆ (𝐷 ↑m 𝐵))
35		ovex 7434	. . . . . . 7 ⊢ (𝐷 ↑m 𝐵) ∈ V
36	35	ssex 5311	. . . . . 6 ⊢ ({𝑓 ∣ 𝑓:𝐵–1-1-onto→𝐷} ⊆ (𝐷 ↑m 𝐵) → {𝑓 ∣ 𝑓:𝐵–1-1-onto→𝐷} ∈ V)
37	34, 36	syl 17	. . . . 5 ⊢ ((𝑔:𝐴–1-1-onto→𝐵 ∧ ℎ:𝐶–1-1-onto→𝐷) → {𝑓 ∣ 𝑓:𝐵–1-1-onto→𝐷} ∈ V)
38		f1oco 6846	. . . . . . . . 9 ⊢ ((ℎ:𝐶–1-1-onto→𝐷 ∧ 𝑥:𝐴–1-1-onto→𝐶) → (ℎ ∘ 𝑥):𝐴–1-1-onto→𝐷)
39	38	adantll 711	. . . . . . . 8 ⊢ (((𝑔:𝐴–1-1-onto→𝐵 ∧ ℎ:𝐶–1-1-onto→𝐷) ∧ 𝑥:𝐴–1-1-onto→𝐶) → (ℎ ∘ 𝑥):𝐴–1-1-onto→𝐷)
40		f1ocnv 6835	. . . . . . . . 9 ⊢ (𝑔:𝐴–1-1-onto→𝐵 → ◡𝑔:𝐵–1-1-onto→𝐴)
41	40	ad2antrr 723	. . . . . . . 8 ⊢ (((𝑔:𝐴–1-1-onto→𝐵 ∧ ℎ:𝐶–1-1-onto→𝐷) ∧ 𝑥:𝐴–1-1-onto→𝐶) → ◡𝑔:𝐵–1-1-onto→𝐴)
42		f1oco 6846	. . . . . . . 8 ⊢ (((ℎ ∘ 𝑥):𝐴–1-1-onto→𝐷 ∧ ◡𝑔:𝐵–1-1-onto→𝐴) → ((ℎ ∘ 𝑥) ∘ ◡𝑔):𝐵–1-1-onto→𝐷)
43	39, 41, 42	syl2anc 583	. . . . . . 7 ⊢ (((𝑔:𝐴–1-1-onto→𝐵 ∧ ℎ:𝐶–1-1-onto→𝐷) ∧ 𝑥:𝐴–1-1-onto→𝐶) → ((ℎ ∘ 𝑥) ∘ ◡𝑔):𝐵–1-1-onto→𝐷)
44	43	ex 412	. . . . . 6 ⊢ ((𝑔:𝐴–1-1-onto→𝐵 ∧ ℎ:𝐶–1-1-onto→𝐷) → (𝑥:𝐴–1-1-onto→𝐶 → ((ℎ ∘ 𝑥) ∘ ◡𝑔):𝐵–1-1-onto→𝐷))
45		vex 3470	. . . . . . 7 ⊢ 𝑥 ∈ V
46		f1oeq1 6811	. . . . . . 7 ⊢ (𝑓 = 𝑥 → (𝑓:𝐴–1-1-onto→𝐶 ↔ 𝑥:𝐴–1-1-onto→𝐶))
47	45, 46	elab 3660	. . . . . 6 ⊢ (𝑥 ∈ {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐶} ↔ 𝑥:𝐴–1-1-onto→𝐶)
48	6, 45	coex 7914	. . . . . . . 8 ⊢ (ℎ ∘ 𝑥) ∈ V
49	10	cnvex 7909	. . . . . . . 8 ⊢ ◡𝑔 ∈ V
50	48, 49	coex 7914	. . . . . . 7 ⊢ ((ℎ ∘ 𝑥) ∘ ◡𝑔) ∈ V
51		f1oeq1 6811	. . . . . . 7 ⊢ (𝑓 = ((ℎ ∘ 𝑥) ∘ ◡𝑔) → (𝑓:𝐵–1-1-onto→𝐷 ↔ ((ℎ ∘ 𝑥) ∘ ◡𝑔):𝐵–1-1-onto→𝐷))
52	50, 51	elab 3660	. . . . . 6 ⊢ (((ℎ ∘ 𝑥) ∘ ◡𝑔) ∈ {𝑓 ∣ 𝑓:𝐵–1-1-onto→𝐷} ↔ ((ℎ ∘ 𝑥) ∘ ◡𝑔):𝐵–1-1-onto→𝐷)
53	44, 47, 52	3imtr4g 296	. . . . 5 ⊢ ((𝑔:𝐴–1-1-onto→𝐵 ∧ ℎ:𝐶–1-1-onto→𝐷) → (𝑥 ∈ {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐶} → ((ℎ ∘ 𝑥) ∘ ◡𝑔) ∈ {𝑓 ∣ 𝑓:𝐵–1-1-onto→𝐷}))
54		f1ocnv 6835	. . . . . . . . 9 ⊢ (ℎ:𝐶–1-1-onto→𝐷 → ◡ℎ:𝐷–1-1-onto→𝐶)
55	54	ad2antlr 724	. . . . . . . 8 ⊢ (((𝑔:𝐴–1-1-onto→𝐵 ∧ ℎ:𝐶–1-1-onto→𝐷) ∧ 𝑦:𝐵–1-1-onto→𝐷) → ◡ℎ:𝐷–1-1-onto→𝐶)
56		f1oco 6846	. . . . . . . . . 10 ⊢ ((𝑦:𝐵–1-1-onto→𝐷 ∧ 𝑔:𝐴–1-1-onto→𝐵) → (𝑦 ∘ 𝑔):𝐴–1-1-onto→𝐷)
57	56	ancoms 458	. . . . . . . . 9 ⊢ ((𝑔:𝐴–1-1-onto→𝐵 ∧ 𝑦:𝐵–1-1-onto→𝐷) → (𝑦 ∘ 𝑔):𝐴–1-1-onto→𝐷)
58	57	adantlr 712	. . . . . . . 8 ⊢ (((𝑔:𝐴–1-1-onto→𝐵 ∧ ℎ:𝐶–1-1-onto→𝐷) ∧ 𝑦:𝐵–1-1-onto→𝐷) → (𝑦 ∘ 𝑔):𝐴–1-1-onto→𝐷)
59		f1oco 6846	. . . . . . . 8 ⊢ ((◡ℎ:𝐷–1-1-onto→𝐶 ∧ (𝑦 ∘ 𝑔):𝐴–1-1-onto→𝐷) → (◡ℎ ∘ (𝑦 ∘ 𝑔)):𝐴–1-1-onto→𝐶)
60	55, 58, 59	syl2anc 583	. . . . . . 7 ⊢ (((𝑔:𝐴–1-1-onto→𝐵 ∧ ℎ:𝐶–1-1-onto→𝐷) ∧ 𝑦:𝐵–1-1-onto→𝐷) → (◡ℎ ∘ (𝑦 ∘ 𝑔)):𝐴–1-1-onto→𝐶)
61	60	ex 412	. . . . . 6 ⊢ ((𝑔:𝐴–1-1-onto→𝐵 ∧ ℎ:𝐶–1-1-onto→𝐷) → (𝑦:𝐵–1-1-onto→𝐷 → (◡ℎ ∘ (𝑦 ∘ 𝑔)):𝐴–1-1-onto→𝐶))
62		vex 3470	. . . . . . 7 ⊢ 𝑦 ∈ V
63		f1oeq1 6811	. . . . . . 7 ⊢ (𝑓 = 𝑦 → (𝑓:𝐵–1-1-onto→𝐷 ↔ 𝑦:𝐵–1-1-onto→𝐷))
64	62, 63	elab 3660	. . . . . 6 ⊢ (𝑦 ∈ {𝑓 ∣ 𝑓:𝐵–1-1-onto→𝐷} ↔ 𝑦:𝐵–1-1-onto→𝐷)
65	6	cnvex 7909	. . . . . . . 8 ⊢ ◡ℎ ∈ V
66	62, 10	coex 7914	. . . . . . . 8 ⊢ (𝑦 ∘ 𝑔) ∈ V
67	65, 66	coex 7914	. . . . . . 7 ⊢ (◡ℎ ∘ (𝑦 ∘ 𝑔)) ∈ V
68		f1oeq1 6811	. . . . . . 7 ⊢ (𝑓 = (◡ℎ ∘ (𝑦 ∘ 𝑔)) → (𝑓:𝐴–1-1-onto→𝐶 ↔ (◡ℎ ∘ (𝑦 ∘ 𝑔)):𝐴–1-1-onto→𝐶))
69	67, 68	elab 3660	. . . . . 6 ⊢ ((◡ℎ ∘ (𝑦 ∘ 𝑔)) ∈ {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐶} ↔ (◡ℎ ∘ (𝑦 ∘ 𝑔)):𝐴–1-1-onto→𝐶)
70	61, 64, 69	3imtr4g 296	. . . . 5 ⊢ ((𝑔:𝐴–1-1-onto→𝐵 ∧ ℎ:𝐶–1-1-onto→𝐷) → (𝑦 ∈ {𝑓 ∣ 𝑓:𝐵–1-1-onto→𝐷} → (◡ℎ ∘ (𝑦 ∘ 𝑔)) ∈ {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐶}))
71	47, 64	anbi12i 626	. . . . . 6 ⊢ ((𝑥 ∈ {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐶} ∧ 𝑦 ∈ {𝑓 ∣ 𝑓:𝐵–1-1-onto→𝐷}) ↔ (𝑥:𝐴–1-1-onto→𝐶 ∧ 𝑦:𝐵–1-1-onto→𝐷))
72		coass 6254	. . . . . . . . . . 11 ⊢ (((ℎ ∘ 𝑥) ∘ ◡𝑔) ∘ 𝑔) = ((ℎ ∘ 𝑥) ∘ (◡𝑔 ∘ 𝑔))
73		f1ococnv1 6852	. . . . . . . . . . . . . 14 ⊢ (𝑔:𝐴–1-1-onto→𝐵 → (◡𝑔 ∘ 𝑔) = ( I ↾ 𝐴))
74	73	ad2antrr 723	. . . . . . . . . . . . 13 ⊢ (((𝑔:𝐴–1-1-onto→𝐵 ∧ ℎ:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐴–1-1-onto→𝐶 ∧ 𝑦:𝐵–1-1-onto→𝐷)) → (◡𝑔 ∘ 𝑔) = ( I ↾ 𝐴))
75	74	coeq2d 5852	. . . . . . . . . . . 12 ⊢ (((𝑔:𝐴–1-1-onto→𝐵 ∧ ℎ:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐴–1-1-onto→𝐶 ∧ 𝑦:𝐵–1-1-onto→𝐷)) → ((ℎ ∘ 𝑥) ∘ (◡𝑔 ∘ 𝑔)) = ((ℎ ∘ 𝑥) ∘ ( I ↾ 𝐴)))
76	39	adantrr 714	. . . . . . . . . . . . 13 ⊢ (((𝑔:𝐴–1-1-onto→𝐵 ∧ ℎ:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐴–1-1-onto→𝐶 ∧ 𝑦:𝐵–1-1-onto→𝐷)) → (ℎ ∘ 𝑥):𝐴–1-1-onto→𝐷)
77		f1of 6823	. . . . . . . . . . . . 13 ⊢ ((ℎ ∘ 𝑥):𝐴–1-1-onto→𝐷 → (ℎ ∘ 𝑥):𝐴⟶𝐷)
78		fcoi1 6755	. . . . . . . . . . . . 13 ⊢ ((ℎ ∘ 𝑥):𝐴⟶𝐷 → ((ℎ ∘ 𝑥) ∘ ( I ↾ 𝐴)) = (ℎ ∘ 𝑥))
79	76, 77, 78	3syl 18	. . . . . . . . . . . 12 ⊢ (((𝑔:𝐴–1-1-onto→𝐵 ∧ ℎ:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐴–1-1-onto→𝐶 ∧ 𝑦:𝐵–1-1-onto→𝐷)) → ((ℎ ∘ 𝑥) ∘ ( I ↾ 𝐴)) = (ℎ ∘ 𝑥))
80	75, 79	eqtrd 2764	. . . . . . . . . . 11 ⊢ (((𝑔:𝐴–1-1-onto→𝐵 ∧ ℎ:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐴–1-1-onto→𝐶 ∧ 𝑦:𝐵–1-1-onto→𝐷)) → ((ℎ ∘ 𝑥) ∘ (◡𝑔 ∘ 𝑔)) = (ℎ ∘ 𝑥))
81	72, 80	eqtr2id 2777	. . . . . . . . . 10 ⊢ (((𝑔:𝐴–1-1-onto→𝐵 ∧ ℎ:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐴–1-1-onto→𝐶 ∧ 𝑦:𝐵–1-1-onto→𝐷)) → (ℎ ∘ 𝑥) = (((ℎ ∘ 𝑥) ∘ ◡𝑔) ∘ 𝑔))
82		coass 6254	. . . . . . . . . . 11 ⊢ ((ℎ ∘ ◡ℎ) ∘ (𝑦 ∘ 𝑔)) = (ℎ ∘ (◡ℎ ∘ (𝑦 ∘ 𝑔)))
83		f1ococnv2 6850	. . . . . . . . . . . . . 14 ⊢ (ℎ:𝐶–1-1-onto→𝐷 → (ℎ ∘ ◡ℎ) = ( I ↾ 𝐷))
84	83	ad2antlr 724	. . . . . . . . . . . . 13 ⊢ (((𝑔:𝐴–1-1-onto→𝐵 ∧ ℎ:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐴–1-1-onto→𝐶 ∧ 𝑦:𝐵–1-1-onto→𝐷)) → (ℎ ∘ ◡ℎ) = ( I ↾ 𝐷))
85	84	coeq1d 5851	. . . . . . . . . . . 12 ⊢ (((𝑔:𝐴–1-1-onto→𝐵 ∧ ℎ:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐴–1-1-onto→𝐶 ∧ 𝑦:𝐵–1-1-onto→𝐷)) → ((ℎ ∘ ◡ℎ) ∘ (𝑦 ∘ 𝑔)) = (( I ↾ 𝐷) ∘ (𝑦 ∘ 𝑔)))
86	58	adantrl 713	. . . . . . . . . . . . 13 ⊢ (((𝑔:𝐴–1-1-onto→𝐵 ∧ ℎ:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐴–1-1-onto→𝐶 ∧ 𝑦:𝐵–1-1-onto→𝐷)) → (𝑦 ∘ 𝑔):𝐴–1-1-onto→𝐷)
87		f1of 6823	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∘ 𝑔):𝐴–1-1-onto→𝐷 → (𝑦 ∘ 𝑔):𝐴⟶𝐷)
88		fcoi2 6756	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∘ 𝑔):𝐴⟶𝐷 → (( I ↾ 𝐷) ∘ (𝑦 ∘ 𝑔)) = (𝑦 ∘ 𝑔))
89	86, 87, 88	3syl 18	. . . . . . . . . . . 12 ⊢ (((𝑔:𝐴–1-1-onto→𝐵 ∧ ℎ:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐴–1-1-onto→𝐶 ∧ 𝑦:𝐵–1-1-onto→𝐷)) → (( I ↾ 𝐷) ∘ (𝑦 ∘ 𝑔)) = (𝑦 ∘ 𝑔))
90	85, 89	eqtrd 2764	. . . . . . . . . . 11 ⊢ (((𝑔:𝐴–1-1-onto→𝐵 ∧ ℎ:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐴–1-1-onto→𝐶 ∧ 𝑦:𝐵–1-1-onto→𝐷)) → ((ℎ ∘ ◡ℎ) ∘ (𝑦 ∘ 𝑔)) = (𝑦 ∘ 𝑔))
91	82, 90	eqtr3id 2778	. . . . . . . . . 10 ⊢ (((𝑔:𝐴–1-1-onto→𝐵 ∧ ℎ:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐴–1-1-onto→𝐶 ∧ 𝑦:𝐵–1-1-onto→𝐷)) → (ℎ ∘ (◡ℎ ∘ (𝑦 ∘ 𝑔))) = (𝑦 ∘ 𝑔))
92	81, 91	eqeq12d 2740	. . . . . . . . 9 ⊢ (((𝑔:𝐴–1-1-onto→𝐵 ∧ ℎ:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐴–1-1-onto→𝐶 ∧ 𝑦:𝐵–1-1-onto→𝐷)) → ((ℎ ∘ 𝑥) = (ℎ ∘ (◡ℎ ∘ (𝑦 ∘ 𝑔))) ↔ (((ℎ ∘ 𝑥) ∘ ◡𝑔) ∘ 𝑔) = (𝑦 ∘ 𝑔)))
93		eqcom 2731	. . . . . . . . 9 ⊢ ((((ℎ ∘ 𝑥) ∘ ◡𝑔) ∘ 𝑔) = (𝑦 ∘ 𝑔) ↔ (𝑦 ∘ 𝑔) = (((ℎ ∘ 𝑥) ∘ ◡𝑔) ∘ 𝑔))
94	92, 93	bitrdi 287	. . . . . . . 8 ⊢ (((𝑔:𝐴–1-1-onto→𝐵 ∧ ℎ:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐴–1-1-onto→𝐶 ∧ 𝑦:𝐵–1-1-onto→𝐷)) → ((ℎ ∘ 𝑥) = (ℎ ∘ (◡ℎ ∘ (𝑦 ∘ 𝑔))) ↔ (𝑦 ∘ 𝑔) = (((ℎ ∘ 𝑥) ∘ ◡𝑔) ∘ 𝑔)))
95		f1of1 6822	. . . . . . . . . 10 ⊢ (ℎ:𝐶–1-1-onto→𝐷 → ℎ:𝐶–1-1→𝐷)
96	95	ad2antlr 724	. . . . . . . . 9 ⊢ (((𝑔:𝐴–1-1-onto→𝐵 ∧ ℎ:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐴–1-1-onto→𝐶 ∧ 𝑦:𝐵–1-1-onto→𝐷)) → ℎ:𝐶–1-1→𝐷)
97		f1of 6823	. . . . . . . . . 10 ⊢ (𝑥:𝐴–1-1-onto→𝐶 → 𝑥:𝐴⟶𝐶)
98	97	ad2antrl 725	. . . . . . . . 9 ⊢ (((𝑔:𝐴–1-1-onto→𝐵 ∧ ℎ:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐴–1-1-onto→𝐶 ∧ 𝑦:𝐵–1-1-onto→𝐷)) → 𝑥:𝐴⟶𝐶)
99	60	adantrl 713	. . . . . . . . . 10 ⊢ (((𝑔:𝐴–1-1-onto→𝐵 ∧ ℎ:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐴–1-1-onto→𝐶 ∧ 𝑦:𝐵–1-1-onto→𝐷)) → (◡ℎ ∘ (𝑦 ∘ 𝑔)):𝐴–1-1-onto→𝐶)
100		f1of 6823	. . . . . . . . . 10 ⊢ ((◡ℎ ∘ (𝑦 ∘ 𝑔)):𝐴–1-1-onto→𝐶 → (◡ℎ ∘ (𝑦 ∘ 𝑔)):𝐴⟶𝐶)
101	99, 100	syl 17	. . . . . . . . 9 ⊢ (((𝑔:𝐴–1-1-onto→𝐵 ∧ ℎ:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐴–1-1-onto→𝐶 ∧ 𝑦:𝐵–1-1-onto→𝐷)) → (◡ℎ ∘ (𝑦 ∘ 𝑔)):𝐴⟶𝐶)
102		cocan1 7281	. . . . . . . . 9 ⊢ ((ℎ:𝐶–1-1→𝐷 ∧ 𝑥:𝐴⟶𝐶 ∧ (◡ℎ ∘ (𝑦 ∘ 𝑔)):𝐴⟶𝐶) → ((ℎ ∘ 𝑥) = (ℎ ∘ (◡ℎ ∘ (𝑦 ∘ 𝑔))) ↔ 𝑥 = (◡ℎ ∘ (𝑦 ∘ 𝑔))))
103	96, 98, 101, 102	syl3anc 1368	. . . . . . . 8 ⊢ (((𝑔:𝐴–1-1-onto→𝐵 ∧ ℎ:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐴–1-1-onto→𝐶 ∧ 𝑦:𝐵–1-1-onto→𝐷)) → ((ℎ ∘ 𝑥) = (ℎ ∘ (◡ℎ ∘ (𝑦 ∘ 𝑔))) ↔ 𝑥 = (◡ℎ ∘ (𝑦 ∘ 𝑔))))
104	26	ad2antrr 723	. . . . . . . . 9 ⊢ (((𝑔:𝐴–1-1-onto→𝐵 ∧ ℎ:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐴–1-1-onto→𝐶 ∧ 𝑦:𝐵–1-1-onto→𝐷)) → 𝑔:𝐴–onto→𝐵)
105		f1ofn 6824	. . . . . . . . . 10 ⊢ (𝑦:𝐵–1-1-onto→𝐷 → 𝑦 Fn 𝐵)
106	105	ad2antll 726	. . . . . . . . 9 ⊢ (((𝑔:𝐴–1-1-onto→𝐵 ∧ ℎ:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐴–1-1-onto→𝐶 ∧ 𝑦:𝐵–1-1-onto→𝐷)) → 𝑦 Fn 𝐵)
107	43	adantrr 714	. . . . . . . . . 10 ⊢ (((𝑔:𝐴–1-1-onto→𝐵 ∧ ℎ:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐴–1-1-onto→𝐶 ∧ 𝑦:𝐵–1-1-onto→𝐷)) → ((ℎ ∘ 𝑥) ∘ ◡𝑔):𝐵–1-1-onto→𝐷)
108		f1ofn 6824	. . . . . . . . . 10 ⊢ (((ℎ ∘ 𝑥) ∘ ◡𝑔):𝐵–1-1-onto→𝐷 → ((ℎ ∘ 𝑥) ∘ ◡𝑔) Fn 𝐵)
109	107, 108	syl 17	. . . . . . . . 9 ⊢ (((𝑔:𝐴–1-1-onto→𝐵 ∧ ℎ:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐴–1-1-onto→𝐶 ∧ 𝑦:𝐵–1-1-onto→𝐷)) → ((ℎ ∘ 𝑥) ∘ ◡𝑔) Fn 𝐵)
110		cocan2 7282	. . . . . . . . 9 ⊢ ((𝑔:𝐴–onto→𝐵 ∧ 𝑦 Fn 𝐵 ∧ ((ℎ ∘ 𝑥) ∘ ◡𝑔) Fn 𝐵) → ((𝑦 ∘ 𝑔) = (((ℎ ∘ 𝑥) ∘ ◡𝑔) ∘ 𝑔) ↔ 𝑦 = ((ℎ ∘ 𝑥) ∘ ◡𝑔)))
111	104, 106, 109, 110	syl3anc 1368	. . . . . . . 8 ⊢ (((𝑔:𝐴–1-1-onto→𝐵 ∧ ℎ:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐴–1-1-onto→𝐶 ∧ 𝑦:𝐵–1-1-onto→𝐷)) → ((𝑦 ∘ 𝑔) = (((ℎ ∘ 𝑥) ∘ ◡𝑔) ∘ 𝑔) ↔ 𝑦 = ((ℎ ∘ 𝑥) ∘ ◡𝑔)))
112	94, 103, 111	3bitr3d 309	. . . . . . 7 ⊢ (((𝑔:𝐴–1-1-onto→𝐵 ∧ ℎ:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐴–1-1-onto→𝐶 ∧ 𝑦:𝐵–1-1-onto→𝐷)) → (𝑥 = (◡ℎ ∘ (𝑦 ∘ 𝑔)) ↔ 𝑦 = ((ℎ ∘ 𝑥) ∘ ◡𝑔)))
113	112	ex 412	. . . . . 6 ⊢ ((𝑔:𝐴–1-1-onto→𝐵 ∧ ℎ:𝐶–1-1-onto→𝐷) → ((𝑥:𝐴–1-1-onto→𝐶 ∧ 𝑦:𝐵–1-1-onto→𝐷) → (𝑥 = (◡ℎ ∘ (𝑦 ∘ 𝑔)) ↔ 𝑦 = ((ℎ ∘ 𝑥) ∘ ◡𝑔))))
114	71, 113	biimtrid 241	. . . . 5 ⊢ ((𝑔:𝐴–1-1-onto→𝐵 ∧ ℎ:𝐶–1-1-onto→𝐷) → ((𝑥 ∈ {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐶} ∧ 𝑦 ∈ {𝑓 ∣ 𝑓:𝐵–1-1-onto→𝐷}) → (𝑥 = (◡ℎ ∘ (𝑦 ∘ 𝑔)) ↔ 𝑦 = ((ℎ ∘ 𝑥) ∘ ◡𝑔))))
115	19, 37, 53, 70, 114	en3d 8980	. . . 4 ⊢ ((𝑔:𝐴–1-1-onto→𝐵 ∧ ℎ:𝐶–1-1-onto→𝐷) → {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐶} ≈ {𝑓 ∣ 𝑓:𝐵–1-1-onto→𝐷})
116	115	exlimivv 1927	. . 3 ⊢ (∃𝑔∃ℎ(𝑔:𝐴–1-1-onto→𝐵 ∧ ℎ:𝐶–1-1-onto→𝐷) → {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐶} ≈ {𝑓 ∣ 𝑓:𝐵–1-1-onto→𝐷})
117	3, 116	sylbir 234	. 2 ⊢ ((∃𝑔 𝑔:𝐴–1-1-onto→𝐵 ∧ ∃ℎ ℎ:𝐶–1-1-onto→𝐷) → {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐶} ≈ {𝑓 ∣ 𝑓:𝐵–1-1-onto→𝐷})
118	1, 2, 117	syl2anb 597	1 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐶} ≈ {𝑓 ∣ 𝑓:𝐵–1-1-onto→𝐷})

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533  ∃wex 1773   ∈ wcel 2098  {cab 2701  Vcvv 3466   ⊆ wss 3940   class class class wbr 5138   I cid 5563  ^◡ccnv 5665  dom cdm 5666  ran crn 5667   ↾ cres 5668   ∘ ccom 5670   Fn wfn 6528  ⟶wf 6529  –1-1→wf1 6530  –onto→wfo 6531  –1-1-onto→wf1o 6532  (class class class)co 7401   ↑_m cmap 8815   ≈ cen 8931

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pow 5353  ax-pr 5417  ax-un 7718

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-sbc 3770  df-csb 3886  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-pw 4596  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-opab 5201  df-mpt 5222  df-id 5564  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6485  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7404  df-oprab 7405  df-mpo 7406  df-map 8817  df-en 8935

This theorem is referenced by: (None)