Theorem hashfacen 14640

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⟶1-1onto𝐶} ≈ {𝑓 ∣ 𝑓:𝐵⟶1-1onto𝐷})

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 9021	. 2 ⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑔 𝑔:𝐴⟶1-1onto𝐵)
2		bren 9021	. 2 ⊢ (𝐶 ≈ 𝐷 ↔ ∃ℎ ℎ:𝐶⟶1-1onto𝐷)
3		exdistrv 1990	. . 3 ⊢ (∃𝑔∃ℎ(𝑔:𝐴⟶1-1onto𝐵 ∧ ℎ:𝐶⟶1-1onto𝐷) ↔ (∃𝑔 𝑔:𝐴⟶1-1onto𝐵 ∧ ∃ℎ ℎ:𝐶⟶1-1onto𝐷))
4		f1osetex 8919	. . . . . 6 ⊢ {𝑓 ∣ 𝑓:𝐴⟶1-1onto𝐶} ∈ V
5	4	a1i 11	. . . . 5 ⊢ ((𝑔:𝐴⟶1-1onto𝐵 ∧ ℎ:𝐶⟶1-1onto𝐷) → {𝑓 ∣ 𝑓:𝐴⟶1-1onto𝐶} ∈ V)
6		f1osetex 8919	. . . . . 6 ⊢ {𝑓 ∣ 𝑓:𝐵⟶1-1onto𝐷} ∈ V
7	6	a1i 11	. . . . 5 ⊢ ((𝑔:𝐴⟶1-1onto𝐵 ∧ ℎ:𝐶⟶1-1onto𝐷) → {𝑓 ∣ 𝑓:𝐵⟶1-1onto𝐷} ∈ V)
8		f1oco 6878	. . . . . . . . 9 ⊢ ((ℎ:𝐶⟶1-1onto𝐷 ∧ 𝑥:𝐴⟶1-1onto𝐶) → (ℎ ∘ 𝑥):𝐴⟶1-1onto𝐷)
9	8	adantll 727	. . . . . . . 8 ⊢ (((𝑔:𝐴⟶1-1onto𝐵 ∧ ℎ:𝐶⟶1-1onto𝐷) ∧ 𝑥:𝐴⟶1-1onto𝐶) → (ℎ ∘ 𝑥):𝐴⟶1-1onto𝐷)
10		f1ocnv 6867	. . . . . . . . 9 ⊢ (𝑔:𝐴⟶1-1onto𝐵 → ◡𝑔:𝐵⟶1-1onto𝐴)
11	10	ad2antrr 739	. . . . . . . 8 ⊢ (((𝑔:𝐴⟶1-1onto𝐵 ∧ ℎ:𝐶⟶1-1onto𝐷) ∧ 𝑥:𝐴⟶1-1onto𝐶) → ◡𝑔:𝐵⟶1-1onto𝐴)
12		f1oco 6878	. . . . . . . 8 ⊢ (((ℎ ∘ 𝑥):𝐴⟶1-1onto𝐷 ∧ ◡𝑔:𝐵⟶1-1onto𝐴) → ((ℎ ∘ 𝑥) ∘ ◡𝑔):𝐵⟶1-1onto𝐷)
13	9, 11, 12	syl2anc 596	. . . . . . 7 ⊢ (((𝑔:𝐴⟶1-1onto𝐵 ∧ ℎ:𝐶⟶1-1onto𝐷) ∧ 𝑥:𝐴⟶1-1onto𝐶) → ((ℎ ∘ 𝑥) ∘ ◡𝑔):𝐵⟶1-1onto𝐷)
14	13	ex 418	. . . . . 6 ⊢ ((𝑔:𝐴⟶1-1onto𝐵 ∧ ℎ:𝐶⟶1-1onto𝐷) → (𝑥:𝐴⟶1-1onto𝐶 → ((ℎ ∘ 𝑥) ∘ ◡𝑔):𝐵⟶1-1onto𝐷))
15		vex 3458	. . . . . . 7 ⊢ 𝑥 ∈ V
16		f1oeq1 6842	. . . . . . 7 ⊢ (𝑓 = 𝑥 → (𝑓:𝐴⟶1-1onto𝐶 ↔ 𝑥:𝐴⟶1-1onto𝐶))
17	15, 16	elab 3636	. . . . . 6 ⊢ (𝑥 ∈ {𝑓 ∣ 𝑓:𝐴⟶1-1onto𝐶} ↔ 𝑥:𝐴⟶1-1onto𝐶)
18		vex 3458	. . . . . . . . 9 ⊢ ℎ ∈ V
19	18, 15	coex 7976	. . . . . . . 8 ⊢ (ℎ ∘ 𝑥) ∈ V
20		vex 3458	. . . . . . . . 9 ⊢ 𝑔 ∈ V
21	20	cnvex 7971	. . . . . . . 8 ⊢ ◡𝑔 ∈ V
22	19, 21	coex 7976	. . . . . . 7 ⊢ ((ℎ ∘ 𝑥) ∘ ◡𝑔) ∈ V
23		f1oeq1 6842	. . . . . . 7 ⊢ (𝑓 = ((ℎ ∘ 𝑥) ∘ ◡𝑔) → (𝑓:𝐵⟶1-1onto𝐷 ↔ ((ℎ ∘ 𝑥) ∘ ◡𝑔):𝐵⟶1-1onto𝐷))
24	22, 23	elab 3636	. . . . . 6 ⊢ (((ℎ ∘ 𝑥) ∘ ◡𝑔) ∈ {𝑓 ∣ 𝑓:𝐵⟶1-1onto𝐷} ↔ ((ℎ ∘ 𝑥) ∘ ◡𝑔):𝐵⟶1-1onto𝐷)
25	14, 17, 24	3imtr4g 299	. . . . 5 ⊢ ((𝑔:𝐴⟶1-1onto𝐵 ∧ ℎ:𝐶⟶1-1onto𝐷) → (𝑥 ∈ {𝑓 ∣ 𝑓:𝐴⟶1-1onto𝐶} → ((ℎ ∘ 𝑥) ∘ ◡𝑔) ∈ {𝑓 ∣ 𝑓:𝐵⟶1-1onto𝐷}))
26		f1ocnv 6867	. . . . . . . . 9 ⊢ (ℎ:𝐶⟶1-1onto𝐷 → ◡ℎ:𝐷⟶1-1onto𝐶)
27	26	ad2antlr 740	. . . . . . . 8 ⊢ (((𝑔:𝐴⟶1-1onto𝐵 ∧ ℎ:𝐶⟶1-1onto𝐷) ∧ 𝑦:𝐵⟶1-1onto𝐷) → ◡ℎ:𝐷⟶1-1onto𝐶)
28		f1oco 6878	. . . . . . . . . 10 ⊢ ((𝑦:𝐵⟶1-1onto𝐷 ∧ 𝑔:𝐴⟶1-1onto𝐵) → (𝑦 ∘ 𝑔):𝐴⟶1-1onto𝐷)
29	28	ancoms 464	. . . . . . . . 9 ⊢ ((𝑔:𝐴⟶1-1onto𝐵 ∧ 𝑦:𝐵⟶1-1onto𝐷) → (𝑦 ∘ 𝑔):𝐴⟶1-1onto𝐷)
30	29	adantlr 728	. . . . . . . 8 ⊢ (((𝑔:𝐴⟶1-1onto𝐵 ∧ ℎ:𝐶⟶1-1onto𝐷) ∧ 𝑦:𝐵⟶1-1onto𝐷) → (𝑦 ∘ 𝑔):𝐴⟶1-1onto𝐷)
31		f1oco 6878	. . . . . . . 8 ⊢ ((◡ℎ:𝐷⟶1-1onto𝐶 ∧ (𝑦 ∘ 𝑔):𝐴⟶1-1onto𝐷) → (◡ℎ ∘ (𝑦 ∘ 𝑔)):𝐴⟶1-1onto𝐶)
32	27, 30, 31	syl2anc 596	. . . . . . 7 ⊢ (((𝑔:𝐴⟶1-1onto𝐵 ∧ ℎ:𝐶⟶1-1onto𝐷) ∧ 𝑦:𝐵⟶1-1onto𝐷) → (◡ℎ ∘ (𝑦 ∘ 𝑔)):𝐴⟶1-1onto𝐶)
33	32	ex 418	. . . . . 6 ⊢ ((𝑔:𝐴⟶1-1onto𝐵 ∧ ℎ:𝐶⟶1-1onto𝐷) → (𝑦:𝐵⟶1-1onto𝐷 → (◡ℎ ∘ (𝑦 ∘ 𝑔)):𝐴⟶1-1onto𝐶))
34		vex 3458	. . . . . . 7 ⊢ 𝑦 ∈ V
35		f1oeq1 6842	. . . . . . 7 ⊢ (𝑓 = 𝑦 → (𝑓:𝐵⟶1-1onto𝐷 ↔ 𝑦:𝐵⟶1-1onto𝐷))
36	34, 35	elab 3636	. . . . . 6 ⊢ (𝑦 ∈ {𝑓 ∣ 𝑓:𝐵⟶1-1onto𝐷} ↔ 𝑦:𝐵⟶1-1onto𝐷)
37	18	cnvex 7971	. . . . . . . 8 ⊢ ◡ℎ ∈ V
38	34, 20	coex 7976	. . . . . . . 8 ⊢ (𝑦 ∘ 𝑔) ∈ V
39	37, 38	coex 7976	. . . . . . 7 ⊢ (◡ℎ ∘ (𝑦 ∘ 𝑔)) ∈ V
40		f1oeq1 6842	. . . . . . 7 ⊢ (𝑓 = (◡ℎ ∘ (𝑦 ∘ 𝑔)) → (𝑓:𝐴⟶1-1onto𝐶 ↔ (◡ℎ ∘ (𝑦 ∘ 𝑔)):𝐴⟶1-1onto𝐶))
41	39, 40	elab 3636	. . . . . 6 ⊢ ((◡ℎ ∘ (𝑦 ∘ 𝑔)) ∈ {𝑓 ∣ 𝑓:𝐴⟶1-1onto𝐶} ↔ (◡ℎ ∘ (𝑦 ∘ 𝑔)):𝐴⟶1-1onto𝐶)
42	33, 36, 41	3imtr4g 299	. . . . 5 ⊢ ((𝑔:𝐴⟶1-1onto𝐵 ∧ ℎ:𝐶⟶1-1onto𝐷) → (𝑦 ∈ {𝑓 ∣ 𝑓:𝐵⟶1-1onto𝐷} → (◡ℎ ∘ (𝑦 ∘ 𝑔)) ∈ {𝑓 ∣ 𝑓:𝐴⟶1-1onto𝐶}))
43	17, 36	anbi12i 640	. . . . . 6 ⊢ ((𝑥 ∈ {𝑓 ∣ 𝑓:𝐴⟶1-1onto𝐶} ∧ 𝑦 ∈ {𝑓 ∣ 𝑓:𝐵⟶1-1onto𝐷}) ↔ (𝑥:𝐴⟶1-1onto𝐶 ∧ 𝑦:𝐵⟶1-1onto𝐷))
44		coass 6290	. . . . . . . . . . 11 ⊢ (((ℎ ∘ 𝑥) ∘ ◡𝑔) ∘ 𝑔) = ((ℎ ∘ 𝑥) ∘ (◡𝑔 ∘ 𝑔))
45		f1ococnv1 6885	. . . . . . . . . . . . . 14 ⊢ (𝑔:𝐴⟶1-1onto𝐵 → (◡𝑔 ∘ 𝑔) = ( I ↾ 𝐴))
46	45	ad2antrr 739	. . . . . . . . . . . . 13 ⊢ (((𝑔:𝐴⟶1-1onto𝐵 ∧ ℎ:𝐶⟶1-1onto𝐷) ∧ (𝑥:𝐴⟶1-1onto𝐶 ∧ 𝑦:𝐵⟶1-1onto𝐷)) → (◡𝑔 ∘ 𝑔) = ( I ↾ 𝐴))
47	46	coeq2d 5851	. . . . . . . . . . . 12 ⊢ (((𝑔:𝐴⟶1-1onto𝐵 ∧ ℎ:𝐶⟶1-1onto𝐷) ∧ (𝑥:𝐴⟶1-1onto𝐶 ∧ 𝑦:𝐵⟶1-1onto𝐷)) → ((ℎ ∘ 𝑥) ∘ (◡𝑔 ∘ 𝑔)) = ((ℎ ∘ 𝑥) ∘ ( I ↾ 𝐴)))
48	9	adantrr 730	. . . . . . . . . . . . 13 ⊢ (((𝑔:𝐴⟶1-1onto𝐵 ∧ ℎ:𝐶⟶1-1onto𝐷) ∧ (𝑥:𝐴⟶1-1onto𝐶 ∧ 𝑦:𝐵⟶1-1onto𝐷)) → (ℎ ∘ 𝑥):𝐴⟶1-1onto𝐷)
49		f1of 6854	. . . . . . . . . . . . 13 ⊢ ((ℎ ∘ 𝑥):𝐴⟶1-1onto𝐷 → (ℎ ∘ 𝑥):𝐴⟶𝐷)
50		fcoi1 6786	. . . . . . . . . . . . 13 ⊢ ((ℎ ∘ 𝑥):𝐴⟶𝐷 → ((ℎ ∘ 𝑥) ∘ ( I ↾ 𝐴)) = (ℎ ∘ 𝑥))
51	48, 49, 50	3syl 19	. . . . . . . . . . . 12 ⊢ (((𝑔:𝐴⟶1-1onto𝐵 ∧ ℎ:𝐶⟶1-1onto𝐷) ∧ (𝑥:𝐴⟶1-1onto𝐶 ∧ 𝑦:𝐵⟶1-1onto𝐷)) → ((ℎ ∘ 𝑥) ∘ ( I ↾ 𝐴)) = (ℎ ∘ 𝑥))
52	47, 51	eqtrd 2798	. . . . . . . . . . 11 ⊢ (((𝑔:𝐴⟶1-1onto𝐵 ∧ ℎ:𝐶⟶1-1onto𝐷) ∧ (𝑥:𝐴⟶1-1onto𝐶 ∧ 𝑦:𝐵⟶1-1onto𝐷)) → ((ℎ ∘ 𝑥) ∘ (◡𝑔 ∘ 𝑔)) = (ℎ ∘ 𝑥))
53	44, 52	eqtr2id 2811	. . . . . . . . . 10 ⊢ (((𝑔:𝐴⟶1-1onto𝐵 ∧ ℎ:𝐶⟶1-1onto𝐷) ∧ (𝑥:𝐴⟶1-1onto𝐶 ∧ 𝑦:𝐵⟶1-1onto𝐷)) → (ℎ ∘ 𝑥) = (((ℎ ∘ 𝑥) ∘ ◡𝑔) ∘ 𝑔))
54		coass 6290	. . . . . . . . . . 11 ⊢ ((ℎ ∘ ◡ℎ) ∘ (𝑦 ∘ 𝑔)) = (ℎ ∘ (◡ℎ ∘ (𝑦 ∘ 𝑔)))
55		f1ococnv2 6883	. . . . . . . . . . . . . 14 ⊢ (ℎ:𝐶⟶1-1onto𝐷 → (ℎ ∘ ◡ℎ) = ( I ↾ 𝐷))
56	55	ad2antlr 740	. . . . . . . . . . . . 13 ⊢ (((𝑔:𝐴⟶1-1onto𝐵 ∧ ℎ:𝐶⟶1-1onto𝐷) ∧ (𝑥:𝐴⟶1-1onto𝐶 ∧ 𝑦:𝐵⟶1-1onto𝐷)) → (ℎ ∘ ◡ℎ) = ( I ↾ 𝐷))
57	56	coeq1d 5850	. . . . . . . . . . . 12 ⊢ (((𝑔:𝐴⟶1-1onto𝐵 ∧ ℎ:𝐶⟶1-1onto𝐷) ∧ (𝑥:𝐴⟶1-1onto𝐶 ∧ 𝑦:𝐵⟶1-1onto𝐷)) → ((ℎ ∘ ◡ℎ) ∘ (𝑦 ∘ 𝑔)) = (( I ↾ 𝐷) ∘ (𝑦 ∘ 𝑔)))
58	30	adantrl 729	. . . . . . . . . . . . 13 ⊢ (((𝑔:𝐴⟶1-1onto𝐵 ∧ ℎ:𝐶⟶1-1onto𝐷) ∧ (𝑥:𝐴⟶1-1onto𝐶 ∧ 𝑦:𝐵⟶1-1onto𝐷)) → (𝑦 ∘ 𝑔):𝐴⟶1-1onto𝐷)
59		f1of 6854	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∘ 𝑔):𝐴⟶1-1onto𝐷 → (𝑦 ∘ 𝑔):𝐴⟶𝐷)
60		fcoi2 6787	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∘ 𝑔):𝐴⟶𝐷 → (( I ↾ 𝐷) ∘ (𝑦 ∘ 𝑔)) = (𝑦 ∘ 𝑔))
61	58, 59, 60	3syl 19	. . . . . . . . . . . 12 ⊢ (((𝑔:𝐴⟶1-1onto𝐵 ∧ ℎ:𝐶⟶1-1onto𝐷) ∧ (𝑥:𝐴⟶1-1onto𝐶 ∧ 𝑦:𝐵⟶1-1onto𝐷)) → (( I ↾ 𝐷) ∘ (𝑦 ∘ 𝑔)) = (𝑦 ∘ 𝑔))
62	57, 61	eqtrd 2798	. . . . . . . . . . 11 ⊢ (((𝑔:𝐴⟶1-1onto𝐵 ∧ ℎ:𝐶⟶1-1onto𝐷) ∧ (𝑥:𝐴⟶1-1onto𝐶 ∧ 𝑦:𝐵⟶1-1onto𝐷)) → ((ℎ ∘ ◡ℎ) ∘ (𝑦 ∘ 𝑔)) = (𝑦 ∘ 𝑔))
63	54, 62	eqtr3id 2812	. . . . . . . . . 10 ⊢ (((𝑔:𝐴⟶1-1onto𝐵 ∧ ℎ:𝐶⟶1-1onto𝐷) ∧ (𝑥:𝐴⟶1-1onto𝐶 ∧ 𝑦:𝐵⟶1-1onto𝐷)) → (ℎ ∘ (◡ℎ ∘ (𝑦 ∘ 𝑔))) = (𝑦 ∘ 𝑔))
64	53, 63	eqeq12d 2779	. . . . . . . . 9 ⊢ (((𝑔:𝐴⟶1-1onto𝐵 ∧ ℎ:𝐶⟶1-1onto𝐷) ∧ (𝑥:𝐴⟶1-1onto𝐶 ∧ 𝑦:𝐵⟶1-1onto𝐷)) → ((ℎ ∘ 𝑥) = (ℎ ∘ (◡ℎ ∘ (𝑦 ∘ 𝑔))) ↔ (((ℎ ∘ 𝑥) ∘ ◡𝑔) ∘ 𝑔) = (𝑦 ∘ 𝑔)))
65		eqcom 2770	. . . . . . . . 9 ⊢ ((((ℎ ∘ 𝑥) ∘ ◡𝑔) ∘ 𝑔) = (𝑦 ∘ 𝑔) ↔ (𝑦 ∘ 𝑔) = (((ℎ ∘ 𝑥) ∘ ◡𝑔) ∘ 𝑔))
66	64, 65	bitrdi 290	. . . . . . . 8 ⊢ (((𝑔:𝐴⟶1-1onto𝐵 ∧ ℎ:𝐶⟶1-1onto𝐷) ∧ (𝑥:𝐴⟶1-1onto𝐶 ∧ 𝑦:𝐵⟶1-1onto𝐷)) → ((ℎ ∘ 𝑥) = (ℎ ∘ (◡ℎ ∘ (𝑦 ∘ 𝑔))) ↔ (𝑦 ∘ 𝑔) = (((ℎ ∘ 𝑥) ∘ ◡𝑔) ∘ 𝑔)))
67		f1of1 6853	. . . . . . . . . 10 ⊢ (ℎ:𝐶⟶1-1onto𝐷 → ℎ:𝐶⟶1-1𝐷)
68	67	ad2antlr 740	. . . . . . . . 9 ⊢ (((𝑔:𝐴⟶1-1onto𝐵 ∧ ℎ:𝐶⟶1-1onto𝐷) ∧ (𝑥:𝐴⟶1-1onto𝐶 ∧ 𝑦:𝐵⟶1-1onto𝐷)) → ℎ:𝐶⟶1-1𝐷)
69		f1of 6854	. . . . . . . . . 10 ⊢ (𝑥:𝐴⟶1-1onto𝐶 → 𝑥:𝐴⟶𝐶)
70	69	ad2antrl 741	. . . . . . . . 9 ⊢ (((𝑔:𝐴⟶1-1onto𝐵 ∧ ℎ:𝐶⟶1-1onto𝐷) ∧ (𝑥:𝐴⟶1-1onto𝐶 ∧ 𝑦:𝐵⟶1-1onto𝐷)) → 𝑥:𝐴⟶𝐶)
71	32	adantrl 729	. . . . . . . . . 10 ⊢ (((𝑔:𝐴⟶1-1onto𝐵 ∧ ℎ:𝐶⟶1-1onto𝐷) ∧ (𝑥:𝐴⟶1-1onto𝐶 ∧ 𝑦:𝐵⟶1-1onto𝐷)) → (◡ℎ ∘ (𝑦 ∘ 𝑔)):𝐴⟶1-1onto𝐶)
72		f1of 6854	. . . . . . . . . 10 ⊢ ((◡ℎ ∘ (𝑦 ∘ 𝑔)):𝐴⟶1-1onto𝐶 → (◡ℎ ∘ (𝑦 ∘ 𝑔)):𝐴⟶𝐶)
73	71, 72	syl 18	. . . . . . . . 9 ⊢ (((𝑔:𝐴⟶1-1onto𝐵 ∧ ℎ:𝐶⟶1-1onto𝐷) ∧ (𝑥:𝐴⟶1-1onto𝐶 ∧ 𝑦:𝐵⟶1-1onto𝐷)) → (◡ℎ ∘ (𝑦 ∘ 𝑔)):𝐴⟶𝐶)
74		cocan1 7330	. . . . . . . . 9 ⊢ ((ℎ:𝐶⟶1-1𝐷 ∧ 𝑥:𝐴⟶𝐶 ∧ (◡ℎ ∘ (𝑦 ∘ 𝑔)):𝐴⟶𝐶) → ((ℎ ∘ 𝑥) = (ℎ ∘ (◡ℎ ∘ (𝑦 ∘ 𝑔))) ↔ 𝑥 = (◡ℎ ∘ (𝑦 ∘ 𝑔))))
75	68, 70, 73, 74	syl3anc 1399	. . . . . . . 8 ⊢ (((𝑔:𝐴⟶1-1onto𝐵 ∧ ℎ:𝐶⟶1-1onto𝐷) ∧ (𝑥:𝐴⟶1-1onto𝐶 ∧ 𝑦:𝐵⟶1-1onto𝐷)) → ((ℎ ∘ 𝑥) = (ℎ ∘ (◡ℎ ∘ (𝑦 ∘ 𝑔))) ↔ 𝑥 = (◡ℎ ∘ (𝑦 ∘ 𝑔))))
76		f1ofo 6862	. . . . . . . . . 10 ⊢ (𝑔:𝐴⟶1-1onto𝐵 → 𝑔:𝐴⟶onto𝐵)
77	76	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝑔:𝐴⟶1-1onto𝐵 ∧ ℎ:𝐶⟶1-1onto𝐷) ∧ (𝑥:𝐴⟶1-1onto𝐶 ∧ 𝑦:𝐵⟶1-1onto𝐷)) → 𝑔:𝐴⟶onto𝐵)
78		f1ofn 6855	. . . . . . . . . 10 ⊢ (𝑦:𝐵⟶1-1onto𝐷 → 𝑦 Fn 𝐵)
79	78	ad2antll 742	. . . . . . . . 9 ⊢ (((𝑔:𝐴⟶1-1onto𝐵 ∧ ℎ:𝐶⟶1-1onto𝐷) ∧ (𝑥:𝐴⟶1-1onto𝐶 ∧ 𝑦:𝐵⟶1-1onto𝐷)) → 𝑦 Fn 𝐵)
80	13	adantrr 730	. . . . . . . . . 10 ⊢ (((𝑔:𝐴⟶1-1onto𝐵 ∧ ℎ:𝐶⟶1-1onto𝐷) ∧ (𝑥:𝐴⟶1-1onto𝐶 ∧ 𝑦:𝐵⟶1-1onto𝐷)) → ((ℎ ∘ 𝑥) ∘ ◡𝑔):𝐵⟶1-1onto𝐷)
81		f1ofn 6855	. . . . . . . . . 10 ⊢ (((ℎ ∘ 𝑥) ∘ ◡𝑔):𝐵⟶1-1onto𝐷 → ((ℎ ∘ 𝑥) ∘ ◡𝑔) Fn 𝐵)
82	80, 81	syl 18	. . . . . . . . 9 ⊢ (((𝑔:𝐴⟶1-1onto𝐵 ∧ ℎ:𝐶⟶1-1onto𝐷) ∧ (𝑥:𝐴⟶1-1onto𝐶 ∧ 𝑦:𝐵⟶1-1onto𝐷)) → ((ℎ ∘ 𝑥) ∘ ◡𝑔) Fn 𝐵)
83		cocan2 7331	. . . . . . . . 9 ⊢ ((𝑔:𝐴⟶onto𝐵 ∧ 𝑦 Fn 𝐵 ∧ ((ℎ ∘ 𝑥) ∘ ◡𝑔) Fn 𝐵) → ((𝑦 ∘ 𝑔) = (((ℎ ∘ 𝑥) ∘ ◡𝑔) ∘ 𝑔) ↔ 𝑦 = ((ℎ ∘ 𝑥) ∘ ◡𝑔)))
84	77, 79, 82, 83	syl3anc 1399	. . . . . . . 8 ⊢ (((𝑔:𝐴⟶1-1onto𝐵 ∧ ℎ:𝐶⟶1-1onto𝐷) ∧ (𝑥:𝐴⟶1-1onto𝐶 ∧ 𝑦:𝐵⟶1-1onto𝐷)) → ((𝑦 ∘ 𝑔) = (((ℎ ∘ 𝑥) ∘ ◡𝑔) ∘ 𝑔) ↔ 𝑦 = ((ℎ ∘ 𝑥) ∘ ◡𝑔)))
85	66, 75, 84	3bitr3d 312	. . . . . . 7 ⊢ (((𝑔:𝐴⟶1-1onto𝐵 ∧ ℎ:𝐶⟶1-1onto𝐷) ∧ (𝑥:𝐴⟶1-1onto𝐶 ∧ 𝑦:𝐵⟶1-1onto𝐷)) → (𝑥 = (◡ℎ ∘ (𝑦 ∘ 𝑔)) ↔ 𝑦 = ((ℎ ∘ 𝑥) ∘ ◡𝑔)))
86	85	ex 418	. . . . . 6 ⊢ ((𝑔:𝐴⟶1-1onto𝐵 ∧ ℎ:𝐶⟶1-1onto𝐷) → ((𝑥:𝐴⟶1-1onto𝐶 ∧ 𝑦:𝐵⟶1-1onto𝐷) → (𝑥 = (◡ℎ ∘ (𝑦 ∘ 𝑔)) ↔ 𝑦 = ((ℎ ∘ 𝑥) ∘ ◡𝑔))))
87	43, 86	biimtrid 245	. . . . 5 ⊢ ((𝑔:𝐴⟶1-1onto𝐵 ∧ ℎ:𝐶⟶1-1onto𝐷) → ((𝑥 ∈ {𝑓 ∣ 𝑓:𝐴⟶1-1onto𝐶} ∧ 𝑦 ∈ {𝑓 ∣ 𝑓:𝐵⟶1-1onto𝐷}) → (𝑥 = (◡ℎ ∘ (𝑦 ∘ 𝑔)) ↔ 𝑦 = ((ℎ ∘ 𝑥) ∘ ◡𝑔))))
88	5, 7, 25, 42, 87	en3d 9054	. . . 4 ⊢ ((𝑔:𝐴⟶1-1onto𝐵 ∧ ℎ:𝐶⟶1-1onto𝐷) → {𝑓 ∣ 𝑓:𝐴⟶1-1onto𝐶} ≈ {𝑓 ∣ 𝑓:𝐵⟶1-1onto𝐷})
89	88	exlimivv 1967	. . 3 ⊢ (∃𝑔∃ℎ(𝑔:𝐴⟶1-1onto𝐵 ∧ ℎ:𝐶⟶1-1onto𝐷) → {𝑓 ∣ 𝑓:𝐴⟶1-1onto𝐶} ≈ {𝑓 ∣ 𝑓:𝐵⟶1-1onto𝐷})
90	3, 89	sylbir 238	. 2 ⊢ ((∃𝑔 𝑔:𝐴⟶1-1onto𝐵 ∧ ∃ℎ ℎ:𝐶⟶1-1onto𝐷) → {𝑓 ∣ 𝑓:𝐴⟶1-1onto𝐶} ≈ {𝑓 ∣ 𝑓:𝐵⟶1-1onto𝐷})
91	1, 2, 90	syl2anb 610	1 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → {𝑓 ∣ 𝑓:𝐴⟶1-1onto𝐶} ≈ {𝑓 ∣ 𝑓:𝐵⟶1-1onto𝐷})

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147  {cab 2741  Vcvv 3454   class class class wbr 5108   I cid 5552  ^◡ccnv 5657   ↾ cres 5660   ∘ ccom 5662   Fn wfn 6561  ⟶wf 6562  ⟶1-1wf1 6563  ⟶ontowfo 6564  ⟶1-1ontowf1o 6565   ≈ cen 9008

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-oprab 7455  df-mpo 7456  df-map 8887  df-en 9012

This theorem is used by:  poimirlem9  38815