Theorem hashfacen 14412

Description: The number of bijections between two sets is a cardinal invariant. (Contributed by Mario Carneiro, 21-Jan-2015.) (Proof shortened by AV, 7-Aug-2024.)

Assertion

Ref	Expression
hashfacen	⊢ ((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 hashfacen

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

Step	Hyp	Ref	Expression
1		bren 8948	. 2 ⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑔 𝑔:𝐴–1-1-onto→𝐵)
2		bren 8948	. 2 ⊢ (𝐶 ≈ 𝐷 ↔ ∃ℎ ℎ:𝐶–1-1-onto→𝐷)
3		exdistrv 1959	. . 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		f1osetex 8852	. . . . . 6 ⊢ {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐶} ∈ V
5	4	a1i 11	. . . . 5 ⊢ ((𝑔:𝐴–1-1-onto→𝐵 ∧ ℎ:𝐶–1-1-onto→𝐷) → {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐶} ∈ V)
6		f1osetex 8852	. . . . . 6 ⊢ {𝑓 ∣ 𝑓:𝐵–1-1-onto→𝐷} ∈ V
7	6	a1i 11	. . . . 5 ⊢ ((𝑔:𝐴–1-1-onto→𝐵 ∧ ℎ:𝐶–1-1-onto→𝐷) → {𝑓 ∣ 𝑓:𝐵–1-1-onto→𝐷} ∈ V)
8		f1oco 6856	. . . . . . . . 9 ⊢ ((ℎ:𝐶–1-1-onto→𝐷 ∧ 𝑥:𝐴–1-1-onto→𝐶) → (ℎ ∘ 𝑥):𝐴–1-1-onto→𝐷)
9	8	adantll 712	. . . . . . . 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)
10		f1ocnv 6845	. . . . . . . . 9 ⊢ (𝑔:𝐴–1-1-onto→𝐵 → ◡𝑔:𝐵–1-1-onto→𝐴)
11	10	ad2antrr 724	. . . . . . . 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)
12		f1oco 6856	. . . . . . . 8 ⊢ (((ℎ ∘ 𝑥):𝐴–1-1-onto→𝐷 ∧ ◡𝑔:𝐵–1-1-onto→𝐴) → ((ℎ ∘ 𝑥) ∘ ◡𝑔):𝐵–1-1-onto→𝐷)
13	9, 11, 12	syl2anc 584	. . . . . . 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)
14	13	ex 413	. . . . . 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))
15		vex 3478	. . . . . . 7 ⊢ 𝑥 ∈ V
16		f1oeq1 6821	. . . . . . 7 ⊢ (𝑓 = 𝑥 → (𝑓:𝐴–1-1-onto→𝐶 ↔ 𝑥:𝐴–1-1-onto→𝐶))
17	15, 16	elab 3668	. . . . . 6 ⊢ (𝑥 ∈ {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐶} ↔ 𝑥:𝐴–1-1-onto→𝐶)
18		vex 3478	. . . . . . . . 9 ⊢ ℎ ∈ V
19	18, 15	coex 7920	. . . . . . . 8 ⊢ (ℎ ∘ 𝑥) ∈ V
20		vex 3478	. . . . . . . . 9 ⊢ 𝑔 ∈ V
21	20	cnvex 7915	. . . . . . . 8 ⊢ ◡𝑔 ∈ V
22	19, 21	coex 7920	. . . . . . 7 ⊢ ((ℎ ∘ 𝑥) ∘ ◡𝑔) ∈ V
23		f1oeq1 6821	. . . . . . 7 ⊢ (𝑓 = ((ℎ ∘ 𝑥) ∘ ◡𝑔) → (𝑓:𝐵–1-1-onto→𝐷 ↔ ((ℎ ∘ 𝑥) ∘ ◡𝑔):𝐵–1-1-onto→𝐷))
24	22, 23	elab 3668	. . . . . 6 ⊢ (((ℎ ∘ 𝑥) ∘ ◡𝑔) ∈ {𝑓 ∣ 𝑓:𝐵–1-1-onto→𝐷} ↔ ((ℎ ∘ 𝑥) ∘ ◡𝑔):𝐵–1-1-onto→𝐷)
25	14, 17, 24	3imtr4g 295	. . . . 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}))
26		f1ocnv 6845	. . . . . . . . 9 ⊢ (ℎ:𝐶–1-1-onto→𝐷 → ◡ℎ:𝐷–1-1-onto→𝐶)
27	26	ad2antlr 725	. . . . . . . 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)
28		f1oco 6856	. . . . . . . . . 10 ⊢ ((𝑦:𝐵–1-1-onto→𝐷 ∧ 𝑔:𝐴–1-1-onto→𝐵) → (𝑦 ∘ 𝑔):𝐴–1-1-onto→𝐷)
29	28	ancoms 459	. . . . . . . . 9 ⊢ ((𝑔:𝐴–1-1-onto→𝐵 ∧ 𝑦:𝐵–1-1-onto→𝐷) → (𝑦 ∘ 𝑔):𝐴–1-1-onto→𝐷)
30	29	adantlr 713	. . . . . . . 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)
31		f1oco 6856	. . . . . . . 8 ⊢ ((◡ℎ:𝐷–1-1-onto→𝐶 ∧ (𝑦 ∘ 𝑔):𝐴–1-1-onto→𝐷) → (◡ℎ ∘ (𝑦 ∘ 𝑔)):𝐴–1-1-onto→𝐶)
32	27, 30, 31	syl2anc 584	. . . . . . 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)
33	32	ex 413	. . . . . 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))
34		vex 3478	. . . . . . 7 ⊢ 𝑦 ∈ V
35		f1oeq1 6821	. . . . . . 7 ⊢ (𝑓 = 𝑦 → (𝑓:𝐵–1-1-onto→𝐷 ↔ 𝑦:𝐵–1-1-onto→𝐷))
36	34, 35	elab 3668	. . . . . 6 ⊢ (𝑦 ∈ {𝑓 ∣ 𝑓:𝐵–1-1-onto→𝐷} ↔ 𝑦:𝐵–1-1-onto→𝐷)
37	18	cnvex 7915	. . . . . . . 8 ⊢ ◡ℎ ∈ V
38	34, 20	coex 7920	. . . . . . . 8 ⊢ (𝑦 ∘ 𝑔) ∈ V
39	37, 38	coex 7920	. . . . . . 7 ⊢ (◡ℎ ∘ (𝑦 ∘ 𝑔)) ∈ V
40		f1oeq1 6821	. . . . . . 7 ⊢ (𝑓 = (◡ℎ ∘ (𝑦 ∘ 𝑔)) → (𝑓:𝐴–1-1-onto→𝐶 ↔ (◡ℎ ∘ (𝑦 ∘ 𝑔)):𝐴–1-1-onto→𝐶))
41	39, 40	elab 3668	. . . . . 6 ⊢ ((◡ℎ ∘ (𝑦 ∘ 𝑔)) ∈ {𝑓 ∣ 𝑓:𝐴–1-1-onto→𝐶} ↔ (◡ℎ ∘ (𝑦 ∘ 𝑔)):𝐴–1-1-onto→𝐶)
42	33, 36, 41	3imtr4g 295	. . . . 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}))
43	17, 36	anbi12i 627	. . . . . 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))
44		coass 6264	. . . . . . . . . . 11 ⊢ (((ℎ ∘ 𝑥) ∘ ◡𝑔) ∘ 𝑔) = ((ℎ ∘ 𝑥) ∘ (◡𝑔 ∘ 𝑔))
45		f1ococnv1 6862	. . . . . . . . . . . . . 14 ⊢ (𝑔:𝐴–1-1-onto→𝐵 → (◡𝑔 ∘ 𝑔) = ( I ↾ 𝐴))
46	45	ad2antrr 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)) → (◡𝑔 ∘ 𝑔) = ( I ↾ 𝐴))
47	46	coeq2d 5862	. . . . . . . . . . . 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)))
48	9	adantrr 715	. . . . . . . . . . . . 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)
49		f1of 6833	. . . . . . . . . . . . 13 ⊢ ((ℎ ∘ 𝑥):𝐴–1-1-onto→𝐷 → (ℎ ∘ 𝑥):𝐴⟶𝐷)
50		fcoi1 6765	. . . . . . . . . . . . 13 ⊢ ((ℎ ∘ 𝑥):𝐴⟶𝐷 → ((ℎ ∘ 𝑥) ∘ ( I ↾ 𝐴)) = (ℎ ∘ 𝑥))
51	48, 49, 50	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))
52	47, 51	eqtrd 2772	. . . . . . . . . . 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))
53	44, 52	eqtr2id 2785	. . . . . . . . . 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))
54		coass 6264	. . . . . . . . . . 11 ⊢ ((ℎ ∘ ◡ℎ) ∘ (𝑦 ∘ 𝑔)) = (ℎ ∘ (◡ℎ ∘ (𝑦 ∘ 𝑔)))
55		f1ococnv2 6860	. . . . . . . . . . . . . 14 ⊢ (ℎ:𝐶–1-1-onto→𝐷 → (ℎ ∘ ◡ℎ) = ( I ↾ 𝐷))
56	55	ad2antlr 725	. . . . . . . . . . . . 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))
57	56	coeq1d 5861	. . . . . . . . . . . 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)))
58	30	adantrl 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)) → (𝑦 ∘ 𝑔):𝐴–1-1-onto→𝐷)
59		f1of 6833	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∘ 𝑔):𝐴–1-1-onto→𝐷 → (𝑦 ∘ 𝑔):𝐴⟶𝐷)
60		fcoi2 6766	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∘ 𝑔):𝐴⟶𝐷 → (( I ↾ 𝐷) ∘ (𝑦 ∘ 𝑔)) = (𝑦 ∘ 𝑔))
61	58, 59, 60	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))
62	57, 61	eqtrd 2772	. . . . . . . . . . 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))
63	54, 62	eqtr3id 2786	. . . . . . . . . 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))
64	53, 63	eqeq12d 2748	. . . . . . . . 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)))
65		eqcom 2739	. . . . . . . . 9 ⊢ ((((ℎ ∘ 𝑥) ∘ ◡𝑔) ∘ 𝑔) = (𝑦 ∘ 𝑔) ↔ (𝑦 ∘ 𝑔) = (((ℎ ∘ 𝑥) ∘ ◡𝑔) ∘ 𝑔))
66	64, 65	bitrdi 286	. . . . . . . 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)))
67		f1of1 6832	. . . . . . . . . 10 ⊢ (ℎ:𝐶–1-1-onto→𝐷 → ℎ:𝐶–1-1→𝐷)
68	67	ad2antlr 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)) → ℎ:𝐶–1-1→𝐷)
69		f1of 6833	. . . . . . . . . 10 ⊢ (𝑥:𝐴–1-1-onto→𝐶 → 𝑥:𝐴⟶𝐶)
70	69	ad2antrl 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)) → 𝑥:𝐴⟶𝐶)
71	32	adantrl 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 ∘ (𝑦 ∘ 𝑔)):𝐴–1-1-onto→𝐶)
72		f1of 6833	. . . . . . . . . 10 ⊢ ((◡ℎ ∘ (𝑦 ∘ 𝑔)):𝐴–1-1-onto→𝐶 → (◡ℎ ∘ (𝑦 ∘ 𝑔)):𝐴⟶𝐶)
73	71, 72	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)
74		cocan1 7288	. . . . . . . . 9 ⊢ ((ℎ:𝐶–1-1→𝐷 ∧ 𝑥:𝐴⟶𝐶 ∧ (◡ℎ ∘ (𝑦 ∘ 𝑔)):𝐴⟶𝐶) → ((ℎ ∘ 𝑥) = (ℎ ∘ (◡ℎ ∘ (𝑦 ∘ 𝑔))) ↔ 𝑥 = (◡ℎ ∘ (𝑦 ∘ 𝑔))))
75	68, 70, 73, 74	syl3anc 1371	. . . . . . . 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))))
76		f1ofo 6840	. . . . . . . . . 10 ⊢ (𝑔:𝐴–1-1-onto→𝐵 → 𝑔:𝐴–onto→𝐵)
77	76	ad2antrr 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)) → 𝑔:𝐴–onto→𝐵)
78		f1ofn 6834	. . . . . . . . . 10 ⊢ (𝑦:𝐵–1-1-onto→𝐷 → 𝑦 Fn 𝐵)
79	78	ad2antll 727	. . . . . . . . 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)
80	13	adantrr 715	. . . . . . . . . 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)
81		f1ofn 6834	. . . . . . . . . 10 ⊢ (((ℎ ∘ 𝑥) ∘ ◡𝑔):𝐵–1-1-onto→𝐷 → ((ℎ ∘ 𝑥) ∘ ◡𝑔) Fn 𝐵)
82	80, 81	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)
83		cocan2 7289	. . . . . . . . 9 ⊢ ((𝑔:𝐴–onto→𝐵 ∧ 𝑦 Fn 𝐵 ∧ ((ℎ ∘ 𝑥) ∘ ◡𝑔) Fn 𝐵) → ((𝑦 ∘ 𝑔) = (((ℎ ∘ 𝑥) ∘ ◡𝑔) ∘ 𝑔) ↔ 𝑦 = ((ℎ ∘ 𝑥) ∘ ◡𝑔)))
84	77, 79, 82, 83	syl3anc 1371	. . . . . . . 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)))
85	66, 75, 84	3bitr3d 308	. . . . . . 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)))
86	85	ex 413	. . . . . 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))))
87	43, 86	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))))
88	5, 7, 25, 42, 87	en3d 8984	. . . 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})
89	88	exlimivv 1935	. . 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})
90	3, 89	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})
91	1, 2, 90	syl2anb 598	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 396   = wceq 1541  ∃wex 1781   ∈ wcel 2106  {cab 2709  Vcvv 3474   class class class wbr 5148   I cid 5573  ^◡ccnv 5675   ↾ cres 5678   ∘ ccom 5680   Fn wfn 6538  ⟶wf 6539  –1-1→wf1 6540  –onto→wfo 6541  –1-1-onto→wf1o 6542   ≈ cen 8935

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-map 8821  df-en 8939

This theorem is referenced by:  poimirlem9  36492