Theorem mapen 7112

Description: Two set exponentiations are equinumerous when their bases and exponents are equinumerous. Theorem 6H(c) of [Enderton] p. 139. (Contributed by NM, 16-Dec-2003.) (Proof shortened by Mario Carneiro, 26-Apr-2015.)

Assertion

Ref	Expression
mapen	⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 ↑𝑚 𝐶) ≈ (𝐵 ↑𝑚 𝐷))

Proof of Theorem mapen

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

Step	Hyp	Ref	Expression
1		bren 6996	. 2 ⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵)
2		bren 6996	. 2 ⊢ (𝐶 ≈ 𝐷 ↔ ∃𝑔 𝑔:𝐶–1-1-onto→𝐷)
3		eeanv 1988	. . 3 ⊢ (∃𝑓∃𝑔(𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ↔ (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 ∧ ∃𝑔 𝑔:𝐶–1-1-onto→𝐷))
4		fnmap 6902	. . . . . 6 ⊢ ↑𝑚 Fn (V × V)
5		f1odm 5623	. . . . . . . 8 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → dom 𝑓 = 𝐴)
6	5	adantr 276	. . . . . . 7 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → dom 𝑓 = 𝐴)
7		vex 2818	. . . . . . . 8 ⊢ 𝑓 ∈ V
8	7	dmex 5029	. . . . . . 7 ⊢ dom 𝑓 ∈ V
9	6, 8	eqeltrrdi 2326	. . . . . 6 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → 𝐴 ∈ V)
10		f1odm 5623	. . . . . . . 8 ⊢ (𝑔:𝐶–1-1-onto→𝐷 → dom 𝑔 = 𝐶)
11	10	adantl 277	. . . . . . 7 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → dom 𝑔 = 𝐶)
12		vex 2818	. . . . . . . 8 ⊢ 𝑔 ∈ V
13	12	dmex 5029	. . . . . . 7 ⊢ dom 𝑔 ∈ V
14	11, 13	eqeltrrdi 2326	. . . . . 6 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → 𝐶 ∈ V)
15		fnovex 6091	. . . . . 6 ⊢ (( ↑𝑚 Fn (V × V) ∧ 𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴 ↑𝑚 𝐶) ∈ V)
16	4, 9, 14, 15	mp3an2i 1379	. . . . 5 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → (𝐴 ↑𝑚 𝐶) ∈ V)
17		f1ofo 5626	. . . . . . . . 9 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴–onto→𝐵)
18	17	adantr 276	. . . . . . . 8 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → 𝑓:𝐴–onto→𝐵)
19		forn 5598	. . . . . . . 8 ⊢ (𝑓:𝐴–onto→𝐵 → ran 𝑓 = 𝐵)
20	18, 19	syl 14	. . . . . . 7 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → ran 𝑓 = 𝐵)
21	7	rnex 5030	. . . . . . 7 ⊢ ran 𝑓 ∈ V
22	20, 21	eqeltrrdi 2326	. . . . . 6 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → 𝐵 ∈ V)
23		f1ofo 5626	. . . . . . . . 9 ⊢ (𝑔:𝐶–1-1-onto→𝐷 → 𝑔:𝐶–onto→𝐷)
24	23	adantl 277	. . . . . . . 8 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → 𝑔:𝐶–onto→𝐷)
25		forn 5598	. . . . . . . 8 ⊢ (𝑔:𝐶–onto→𝐷 → ran 𝑔 = 𝐷)
26	24, 25	syl 14	. . . . . . 7 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → ran 𝑔 = 𝐷)
27	12	rnex 5030	. . . . . . 7 ⊢ ran 𝑔 ∈ V
28	26, 27	eqeltrrdi 2326	. . . . . 6 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → 𝐷 ∈ V)
29		fnovex 6091	. . . . . 6 ⊢ (( ↑𝑚 Fn (V × V) ∧ 𝐵 ∈ V ∧ 𝐷 ∈ V) → (𝐵 ↑𝑚 𝐷) ∈ V)
30	4, 22, 28, 29	mp3an2i 1379	. . . . 5 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → (𝐵 ↑𝑚 𝐷) ∈ V)
31		elmapi 6917	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ↑𝑚 𝐶) → 𝑥:𝐶⟶𝐴)
32		f1of 5619	. . . . . . . . . . 11 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴⟶𝐵)
33	32	adantr 276	. . . . . . . . . 10 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → 𝑓:𝐴⟶𝐵)
34		fco 5532	. . . . . . . . . 10 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥:𝐶⟶𝐴) → (𝑓 ∘ 𝑥):𝐶⟶𝐵)
35	33, 34	sylan 283	. . . . . . . . 9 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ 𝑥:𝐶⟶𝐴) → (𝑓 ∘ 𝑥):𝐶⟶𝐵)
36		f1ocnv 5632	. . . . . . . . . . . 12 ⊢ (𝑔:𝐶–1-1-onto→𝐷 → ◡𝑔:𝐷–1-1-onto→𝐶)
37	36	adantl 277	. . . . . . . . . . 11 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → ◡𝑔:𝐷–1-1-onto→𝐶)
38		f1of 5619	. . . . . . . . . . 11 ⊢ (◡𝑔:𝐷–1-1-onto→𝐶 → ◡𝑔:𝐷⟶𝐶)
39	37, 38	syl 14	. . . . . . . . . 10 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → ◡𝑔:𝐷⟶𝐶)
40	39	adantr 276	. . . . . . . . 9 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ 𝑥:𝐶⟶𝐴) → ◡𝑔:𝐷⟶𝐶)
41	35, 40	fcod 5533	. . . . . . . 8 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ 𝑥:𝐶⟶𝐴) → ((𝑓 ∘ 𝑥) ∘ ◡𝑔):𝐷⟶𝐵)
42	41	ex 115	. . . . . . 7 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → (𝑥:𝐶⟶𝐴 → ((𝑓 ∘ 𝑥) ∘ ◡𝑔):𝐷⟶𝐵))
43	31, 42	syl5 32	. . . . . 6 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → (𝑥 ∈ (𝐴 ↑𝑚 𝐶) → ((𝑓 ∘ 𝑥) ∘ ◡𝑔):𝐷⟶𝐵))
44	22, 28	elmapd 6909	. . . . . 6 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → (((𝑓 ∘ 𝑥) ∘ ◡𝑔) ∈ (𝐵 ↑𝑚 𝐷) ↔ ((𝑓 ∘ 𝑥) ∘ ◡𝑔):𝐷⟶𝐵))
45	43, 44	sylibrd 169	. . . . 5 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → (𝑥 ∈ (𝐴 ↑𝑚 𝐶) → ((𝑓 ∘ 𝑥) ∘ ◡𝑔) ∈ (𝐵 ↑𝑚 𝐷)))
46		elmapi 6917	. . . . . . 7 ⊢ (𝑦 ∈ (𝐵 ↑𝑚 𝐷) → 𝑦:𝐷⟶𝐵)
47		f1ocnv 5632	. . . . . . . . . . 11 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → ◡𝑓:𝐵–1-1-onto→𝐴)
48	47	adantr 276	. . . . . . . . . 10 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → ◡𝑓:𝐵–1-1-onto→𝐴)
49		f1of 5619	. . . . . . . . . 10 ⊢ (◡𝑓:𝐵–1-1-onto→𝐴 → ◡𝑓:𝐵⟶𝐴)
50	48, 49	syl 14	. . . . . . . . 9 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → ◡𝑓:𝐵⟶𝐴)
51		id 19	. . . . . . . . . 10 ⊢ (𝑦:𝐷⟶𝐵 → 𝑦:𝐷⟶𝐵)
52		f1of 5619	. . . . . . . . . . 11 ⊢ (𝑔:𝐶–1-1-onto→𝐷 → 𝑔:𝐶⟶𝐷)
53	52	adantl 277	. . . . . . . . . 10 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → 𝑔:𝐶⟶𝐷)
54		fco 5532	. . . . . . . . . 10 ⊢ ((𝑦:𝐷⟶𝐵 ∧ 𝑔:𝐶⟶𝐷) → (𝑦 ∘ 𝑔):𝐶⟶𝐵)
55	51, 53, 54	syl2anr 290	. . . . . . . . 9 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ 𝑦:𝐷⟶𝐵) → (𝑦 ∘ 𝑔):𝐶⟶𝐵)
56		fco 5532	. . . . . . . . 9 ⊢ ((◡𝑓:𝐵⟶𝐴 ∧ (𝑦 ∘ 𝑔):𝐶⟶𝐵) → (◡𝑓 ∘ (𝑦 ∘ 𝑔)):𝐶⟶𝐴)
57	50, 55, 56	syl2an2r 599	. . . . . . . 8 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ 𝑦:𝐷⟶𝐵) → (◡𝑓 ∘ (𝑦 ∘ 𝑔)):𝐶⟶𝐴)
58	57	ex 115	. . . . . . 7 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → (𝑦:𝐷⟶𝐵 → (◡𝑓 ∘ (𝑦 ∘ 𝑔)):𝐶⟶𝐴))
59	46, 58	syl5 32	. . . . . 6 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → (𝑦 ∈ (𝐵 ↑𝑚 𝐷) → (◡𝑓 ∘ (𝑦 ∘ 𝑔)):𝐶⟶𝐴))
60	9, 14	elmapd 6909	. . . . . 6 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → ((◡𝑓 ∘ (𝑦 ∘ 𝑔)) ∈ (𝐴 ↑𝑚 𝐶) ↔ (◡𝑓 ∘ (𝑦 ∘ 𝑔)):𝐶⟶𝐴))
61	59, 60	sylibrd 169	. . . . 5 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → (𝑦 ∈ (𝐵 ↑𝑚 𝐷) → (◡𝑓 ∘ (𝑦 ∘ 𝑔)) ∈ (𝐴 ↑𝑚 𝐶)))
62		coass 5286	. . . . . . . . . . 11 ⊢ ((𝑓 ∘ ◡𝑓) ∘ (𝑦 ∘ 𝑔)) = (𝑓 ∘ (◡𝑓 ∘ (𝑦 ∘ 𝑔)))
63		f1ococnv2 5646	. . . . . . . . . . . . . 14 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → (𝑓 ∘ ◡𝑓) = ( I ↾ 𝐵))
64	63	ad2antrr 488	. . . . . . . . . . . . 13 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → (𝑓 ∘ ◡𝑓) = ( I ↾ 𝐵))
65	64	coeq1d 4921	. . . . . . . . . . . 12 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → ((𝑓 ∘ ◡𝑓) ∘ (𝑦 ∘ 𝑔)) = (( I ↾ 𝐵) ∘ (𝑦 ∘ 𝑔)))
66	55	adantrl 478	. . . . . . . . . . . . 13 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → (𝑦 ∘ 𝑔):𝐶⟶𝐵)
67		fcoi2 5553	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∘ 𝑔):𝐶⟶𝐵 → (( I ↾ 𝐵) ∘ (𝑦 ∘ 𝑔)) = (𝑦 ∘ 𝑔))
68	66, 67	syl 14	. . . . . . . . . . . 12 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → (( I ↾ 𝐵) ∘ (𝑦 ∘ 𝑔)) = (𝑦 ∘ 𝑔))
69	65, 68	eqtrd 2267	. . . . . . . . . . 11 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → ((𝑓 ∘ ◡𝑓) ∘ (𝑦 ∘ 𝑔)) = (𝑦 ∘ 𝑔))
70	62, 69	eqtr3id 2281	. . . . . . . . . 10 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → (𝑓 ∘ (◡𝑓 ∘ (𝑦 ∘ 𝑔))) = (𝑦 ∘ 𝑔))
71	70	eqeq2d 2246	. . . . . . . . 9 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → ((𝑓 ∘ 𝑥) = (𝑓 ∘ (◡𝑓 ∘ (𝑦 ∘ 𝑔))) ↔ (𝑓 ∘ 𝑥) = (𝑦 ∘ 𝑔)))
72		coass 5286	. . . . . . . . . . . 12 ⊢ (((𝑓 ∘ 𝑥) ∘ ◡𝑔) ∘ 𝑔) = ((𝑓 ∘ 𝑥) ∘ (◡𝑔 ∘ 𝑔))
73		f1ococnv1 5648	. . . . . . . . . . . . . . 15 ⊢ (𝑔:𝐶–1-1-onto→𝐷 → (◡𝑔 ∘ 𝑔) = ( I ↾ 𝐶))
74	73	ad2antlr 489	. . . . . . . . . . . . . 14 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → (◡𝑔 ∘ 𝑔) = ( I ↾ 𝐶))
75	74	coeq2d 4922	. . . . . . . . . . . . 13 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → ((𝑓 ∘ 𝑥) ∘ (◡𝑔 ∘ 𝑔)) = ((𝑓 ∘ 𝑥) ∘ ( I ↾ 𝐶)))
76	35	adantrr 479	. . . . . . . . . . . . . 14 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → (𝑓 ∘ 𝑥):𝐶⟶𝐵)
77		fcoi1 5552	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∘ 𝑥):𝐶⟶𝐵 → ((𝑓 ∘ 𝑥) ∘ ( I ↾ 𝐶)) = (𝑓 ∘ 𝑥))
78	76, 77	syl 14	. . . . . . . . . . . . 13 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → ((𝑓 ∘ 𝑥) ∘ ( I ↾ 𝐶)) = (𝑓 ∘ 𝑥))
79	75, 78	eqtrd 2267	. . . . . . . . . . . 12 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → ((𝑓 ∘ 𝑥) ∘ (◡𝑔 ∘ 𝑔)) = (𝑓 ∘ 𝑥))
80	72, 79	eqtrid 2279	. . . . . . . . . . 11 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → (((𝑓 ∘ 𝑥) ∘ ◡𝑔) ∘ 𝑔) = (𝑓 ∘ 𝑥))
81	80	eqeq2d 2246	. . . . . . . . . 10 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → ((𝑦 ∘ 𝑔) = (((𝑓 ∘ 𝑥) ∘ ◡𝑔) ∘ 𝑔) ↔ (𝑦 ∘ 𝑔) = (𝑓 ∘ 𝑥)))
82		eqcom 2236	. . . . . . . . . 10 ⊢ ((𝑦 ∘ 𝑔) = (𝑓 ∘ 𝑥) ↔ (𝑓 ∘ 𝑥) = (𝑦 ∘ 𝑔))
83	81, 82	bitrdi 196	. . . . . . . . 9 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → ((𝑦 ∘ 𝑔) = (((𝑓 ∘ 𝑥) ∘ ◡𝑔) ∘ 𝑔) ↔ (𝑓 ∘ 𝑥) = (𝑦 ∘ 𝑔)))
84	71, 83	bitr4d 191	. . . . . . . 8 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → ((𝑓 ∘ 𝑥) = (𝑓 ∘ (◡𝑓 ∘ (𝑦 ∘ 𝑔))) ↔ (𝑦 ∘ 𝑔) = (((𝑓 ∘ 𝑥) ∘ ◡𝑔) ∘ 𝑔)))
85		f1of1 5618	. . . . . . . . . 10 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴–1-1→𝐵)
86	85	ad2antrr 488	. . . . . . . . 9 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → 𝑓:𝐴–1-1→𝐵)
87		simprl 531	. . . . . . . . 9 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → 𝑥:𝐶⟶𝐴)
88	57	adantrl 478	. . . . . . . . 9 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → (◡𝑓 ∘ (𝑦 ∘ 𝑔)):𝐶⟶𝐴)
89		cocan1 5966	. . . . . . . . 9 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝑥:𝐶⟶𝐴 ∧ (◡𝑓 ∘ (𝑦 ∘ 𝑔)):𝐶⟶𝐴) → ((𝑓 ∘ 𝑥) = (𝑓 ∘ (◡𝑓 ∘ (𝑦 ∘ 𝑔))) ↔ 𝑥 = (◡𝑓 ∘ (𝑦 ∘ 𝑔))))
90	86, 87, 88, 89	syl3anc 1274	. . . . . . . 8 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → ((𝑓 ∘ 𝑥) = (𝑓 ∘ (◡𝑓 ∘ (𝑦 ∘ 𝑔))) ↔ 𝑥 = (◡𝑓 ∘ (𝑦 ∘ 𝑔))))
91	24	adantr 276	. . . . . . . . 9 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → 𝑔:𝐶–onto→𝐷)
92		ffn 5513	. . . . . . . . . 10 ⊢ (𝑦:𝐷⟶𝐵 → 𝑦 Fn 𝐷)
93	92	ad2antll 491	. . . . . . . . 9 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → 𝑦 Fn 𝐷)
94	41	adantrr 479	. . . . . . . . . 10 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → ((𝑓 ∘ 𝑥) ∘ ◡𝑔):𝐷⟶𝐵)
95	94	ffnd 5514	. . . . . . . . 9 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → ((𝑓 ∘ 𝑥) ∘ ◡𝑔) Fn 𝐷)
96		cocan2 5967	. . . . . . . . 9 ⊢ ((𝑔:𝐶–onto→𝐷 ∧ 𝑦 Fn 𝐷 ∧ ((𝑓 ∘ 𝑥) ∘ ◡𝑔) Fn 𝐷) → ((𝑦 ∘ 𝑔) = (((𝑓 ∘ 𝑥) ∘ ◡𝑔) ∘ 𝑔) ↔ 𝑦 = ((𝑓 ∘ 𝑥) ∘ ◡𝑔)))
97	91, 93, 95, 96	syl3anc 1274	. . . . . . . 8 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → ((𝑦 ∘ 𝑔) = (((𝑓 ∘ 𝑥) ∘ ◡𝑔) ∘ 𝑔) ↔ 𝑦 = ((𝑓 ∘ 𝑥) ∘ ◡𝑔)))
98	84, 90, 97	3bitr3d 218	. . . . . . 7 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → (𝑥 = (◡𝑓 ∘ (𝑦 ∘ 𝑔)) ↔ 𝑦 = ((𝑓 ∘ 𝑥) ∘ ◡𝑔)))
99	98	ex 115	. . . . . 6 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → ((𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵) → (𝑥 = (◡𝑓 ∘ (𝑦 ∘ 𝑔)) ↔ 𝑦 = ((𝑓 ∘ 𝑥) ∘ ◡𝑔))))
100	31, 46, 99	syl2ani 408	. . . . 5 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → ((𝑥 ∈ (𝐴 ↑𝑚 𝐶) ∧ 𝑦 ∈ (𝐵 ↑𝑚 𝐷)) → (𝑥 = (◡𝑓 ∘ (𝑦 ∘ 𝑔)) ↔ 𝑦 = ((𝑓 ∘ 𝑥) ∘ ◡𝑔))))
101	16, 30, 45, 61, 100	en3d 7021	. . . 4 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → (𝐴 ↑𝑚 𝐶) ≈ (𝐵 ↑𝑚 𝐷))
102	101	exlimivv 1948	. . 3 ⊢ (∃𝑓∃𝑔(𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → (𝐴 ↑𝑚 𝐶) ≈ (𝐵 ↑𝑚 𝐷))
103	3, 102	sylbir 135	. 2 ⊢ ((∃𝑓 𝑓:𝐴–1-1-onto→𝐵 ∧ ∃𝑔 𝑔:𝐶–1-1-onto→𝐷) → (𝐴 ↑𝑚 𝐶) ≈ (𝐵 ↑𝑚 𝐷))
104	1, 2, 103	syl2anb 291	1 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 ↑𝑚 𝐶) ≈ (𝐵 ↑𝑚 𝐷))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398  ∃wex 1541   ∈ wcel 2205  Vcvv 2815   class class class wbr 4114   I cid 4414   × cxp 4752  ^◡ccnv 4753  dom cdm 4754  ran crn 4755   ↾ cres 4756   ∘ ccom 4758   Fn wfn 5352  ⟶wf 5353  –1-1→wf1 5354  –onto→wfo 5355  –1-1-onto→wf1o 5356  (class class class)co 6058   ↑_𝑚 cmap 6895   ≈ cen 6986

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-map 6897  df-en 6989

This theorem is referenced by:  mapdom1g  7113