Theorem mapen 6824

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 6725	. 2 ⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵)
2		bren 6725	. 2 ⊢ (𝐶 ≈ 𝐷 ↔ ∃𝑔 𝑔:𝐶–1-1-onto→𝐷)
3		eeanv 1925	. . 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 6633	. . . . . 6 ⊢ ↑𝑚 Fn (V × V)
5		f1odm 5446	. . . . . . . 8 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → dom 𝑓 = 𝐴)
6	5	adantr 274	. . . . . . 7 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → dom 𝑓 = 𝐴)
7		vex 2733	. . . . . . . 8 ⊢ 𝑓 ∈ V
8	7	dmex 4877	. . . . . . 7 ⊢ dom 𝑓 ∈ V
9	6, 8	eqeltrrdi 2262	. . . . . 6 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → 𝐴 ∈ V)
10		f1odm 5446	. . . . . . . 8 ⊢ (𝑔:𝐶–1-1-onto→𝐷 → dom 𝑔 = 𝐶)
11	10	adantl 275	. . . . . . 7 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → dom 𝑔 = 𝐶)
12		vex 2733	. . . . . . . 8 ⊢ 𝑔 ∈ V
13	12	dmex 4877	. . . . . . 7 ⊢ dom 𝑔 ∈ V
14	11, 13	eqeltrrdi 2262	. . . . . 6 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → 𝐶 ∈ V)
15		fnovex 5886	. . . . . 6 ⊢ (( ↑𝑚 Fn (V × V) ∧ 𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴 ↑𝑚 𝐶) ∈ V)
16	4, 9, 14, 15	mp3an2i 1337	. . . . 5 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → (𝐴 ↑𝑚 𝐶) ∈ V)
17		f1ofo 5449	. . . . . . . . 9 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴–onto→𝐵)
18	17	adantr 274	. . . . . . . 8 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → 𝑓:𝐴–onto→𝐵)
19		forn 5423	. . . . . . . 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 4878	. . . . . . 7 ⊢ ran 𝑓 ∈ V
22	20, 21	eqeltrrdi 2262	. . . . . 6 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → 𝐵 ∈ V)
23		f1ofo 5449	. . . . . . . . 9 ⊢ (𝑔:𝐶–1-1-onto→𝐷 → 𝑔:𝐶–onto→𝐷)
24	23	adantl 275	. . . . . . . 8 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → 𝑔:𝐶–onto→𝐷)
25		forn 5423	. . . . . . . 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 4878	. . . . . . 7 ⊢ ran 𝑔 ∈ V
28	26, 27	eqeltrrdi 2262	. . . . . 6 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → 𝐷 ∈ V)
29		fnovex 5886	. . . . . 6 ⊢ (( ↑𝑚 Fn (V × V) ∧ 𝐵 ∈ V ∧ 𝐷 ∈ V) → (𝐵 ↑𝑚 𝐷) ∈ V)
30	4, 22, 28, 29	mp3an2i 1337	. . . . 5 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → (𝐵 ↑𝑚 𝐷) ∈ V)
31		elmapi 6648	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ↑𝑚 𝐶) → 𝑥:𝐶⟶𝐴)
32		f1of 5442	. . . . . . . . . . 11 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴⟶𝐵)
33	32	adantr 274	. . . . . . . . . 10 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → 𝑓:𝐴⟶𝐵)
34		fco 5363	. . . . . . . . . 10 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥:𝐶⟶𝐴) → (𝑓 ∘ 𝑥):𝐶⟶𝐵)
35	33, 34	sylan 281	. . . . . . . . 9 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ 𝑥:𝐶⟶𝐴) → (𝑓 ∘ 𝑥):𝐶⟶𝐵)
36		f1ocnv 5455	. . . . . . . . . . . 12 ⊢ (𝑔:𝐶–1-1-onto→𝐷 → ◡𝑔:𝐷–1-1-onto→𝐶)
37	36	adantl 275	. . . . . . . . . . 11 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → ◡𝑔:𝐷–1-1-onto→𝐶)
38		f1of 5442	. . . . . . . . . . 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 274	. . . . . . . . 9 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ 𝑥:𝐶⟶𝐴) → ◡𝑔:𝐷⟶𝐶)
41		fco 5363	. . . . . . . . 9 ⊢ (((𝑓 ∘ 𝑥):𝐶⟶𝐵 ∧ ◡𝑔:𝐷⟶𝐶) → ((𝑓 ∘ 𝑥) ∘ ◡𝑔):𝐷⟶𝐵)
42	35, 40, 41	syl2anc 409	. . . . . . . 8 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ 𝑥:𝐶⟶𝐴) → ((𝑓 ∘ 𝑥) ∘ ◡𝑔):𝐷⟶𝐵)
43	42	ex 114	. . . . . . 7 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → (𝑥:𝐶⟶𝐴 → ((𝑓 ∘ 𝑥) ∘ ◡𝑔):𝐷⟶𝐵))
44	31, 43	syl5 32	. . . . . 6 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → (𝑥 ∈ (𝐴 ↑𝑚 𝐶) → ((𝑓 ∘ 𝑥) ∘ ◡𝑔):𝐷⟶𝐵))
45	22, 28	elmapd 6640	. . . . . 6 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → (((𝑓 ∘ 𝑥) ∘ ◡𝑔) ∈ (𝐵 ↑𝑚 𝐷) ↔ ((𝑓 ∘ 𝑥) ∘ ◡𝑔):𝐷⟶𝐵))
46	44, 45	sylibrd 168	. . . . 5 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → (𝑥 ∈ (𝐴 ↑𝑚 𝐶) → ((𝑓 ∘ 𝑥) ∘ ◡𝑔) ∈ (𝐵 ↑𝑚 𝐷)))
47		elmapi 6648	. . . . . . 7 ⊢ (𝑦 ∈ (𝐵 ↑𝑚 𝐷) → 𝑦:𝐷⟶𝐵)
48		f1ocnv 5455	. . . . . . . . . . 11 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → ◡𝑓:𝐵–1-1-onto→𝐴)
49	48	adantr 274	. . . . . . . . . 10 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → ◡𝑓:𝐵–1-1-onto→𝐴)
50		f1of 5442	. . . . . . . . . 10 ⊢ (◡𝑓:𝐵–1-1-onto→𝐴 → ◡𝑓:𝐵⟶𝐴)
51	49, 50	syl 14	. . . . . . . . 9 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → ◡𝑓:𝐵⟶𝐴)
52		id 19	. . . . . . . . . 10 ⊢ (𝑦:𝐷⟶𝐵 → 𝑦:𝐷⟶𝐵)
53		f1of 5442	. . . . . . . . . . 11 ⊢ (𝑔:𝐶–1-1-onto→𝐷 → 𝑔:𝐶⟶𝐷)
54	53	adantl 275	. . . . . . . . . 10 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → 𝑔:𝐶⟶𝐷)
55		fco 5363	. . . . . . . . . 10 ⊢ ((𝑦:𝐷⟶𝐵 ∧ 𝑔:𝐶⟶𝐷) → (𝑦 ∘ 𝑔):𝐶⟶𝐵)
56	52, 54, 55	syl2anr 288	. . . . . . . . 9 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ 𝑦:𝐷⟶𝐵) → (𝑦 ∘ 𝑔):𝐶⟶𝐵)
57		fco 5363	. . . . . . . . 9 ⊢ ((◡𝑓:𝐵⟶𝐴 ∧ (𝑦 ∘ 𝑔):𝐶⟶𝐵) → (◡𝑓 ∘ (𝑦 ∘ 𝑔)):𝐶⟶𝐴)
58	51, 56, 57	syl2an2r 590	. . . . . . . 8 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ 𝑦:𝐷⟶𝐵) → (◡𝑓 ∘ (𝑦 ∘ 𝑔)):𝐶⟶𝐴)
59	58	ex 114	. . . . . . 7 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → (𝑦:𝐷⟶𝐵 → (◡𝑓 ∘ (𝑦 ∘ 𝑔)):𝐶⟶𝐴))
60	47, 59	syl5 32	. . . . . 6 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → (𝑦 ∈ (𝐵 ↑𝑚 𝐷) → (◡𝑓 ∘ (𝑦 ∘ 𝑔)):𝐶⟶𝐴))
61	9, 14	elmapd 6640	. . . . . 6 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → ((◡𝑓 ∘ (𝑦 ∘ 𝑔)) ∈ (𝐴 ↑𝑚 𝐶) ↔ (◡𝑓 ∘ (𝑦 ∘ 𝑔)):𝐶⟶𝐴))
62	60, 61	sylibrd 168	. . . . 5 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → (𝑦 ∈ (𝐵 ↑𝑚 𝐷) → (◡𝑓 ∘ (𝑦 ∘ 𝑔)) ∈ (𝐴 ↑𝑚 𝐶)))
63		coass 5129	. . . . . . . . . . 11 ⊢ ((𝑓 ∘ ◡𝑓) ∘ (𝑦 ∘ 𝑔)) = (𝑓 ∘ (◡𝑓 ∘ (𝑦 ∘ 𝑔)))
64		f1ococnv2 5469	. . . . . . . . . . . . . 14 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → (𝑓 ∘ ◡𝑓) = ( I ↾ 𝐵))
65	64	ad2antrr 485	. . . . . . . . . . . . 13 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → (𝑓 ∘ ◡𝑓) = ( I ↾ 𝐵))
66	65	coeq1d 4772	. . . . . . . . . . . 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)))
67	56	adantrl 475	. . . . . . . . . . . . 13 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → (𝑦 ∘ 𝑔):𝐶⟶𝐵)
68		fcoi2 5379	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∘ 𝑔):𝐶⟶𝐵 → (( I ↾ 𝐵) ∘ (𝑦 ∘ 𝑔)) = (𝑦 ∘ 𝑔))
69	67, 68	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))
70	66, 69	eqtrd 2203	. . . . . . . . . . 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))
71	63, 70	eqtr3id 2217	. . . . . . . . . 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))
72	71	eqeq2d 2182	. . . . . . . . 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)))
73		coass 5129	. . . . . . . . . . . 12 ⊢ (((𝑓 ∘ 𝑥) ∘ ◡𝑔) ∘ 𝑔) = ((𝑓 ∘ 𝑥) ∘ (◡𝑔 ∘ 𝑔))
74		f1ococnv1 5471	. . . . . . . . . . . . . . 15 ⊢ (𝑔:𝐶–1-1-onto→𝐷 → (◡𝑔 ∘ 𝑔) = ( I ↾ 𝐶))
75	74	ad2antlr 486	. . . . . . . . . . . . . 14 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → (◡𝑔 ∘ 𝑔) = ( I ↾ 𝐶))
76	75	coeq2d 4773	. . . . . . . . . . . . 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)))
77	35	adantrr 476	. . . . . . . . . . . . . 14 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → (𝑓 ∘ 𝑥):𝐶⟶𝐵)
78		fcoi1 5378	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∘ 𝑥):𝐶⟶𝐵 → ((𝑓 ∘ 𝑥) ∘ ( I ↾ 𝐶)) = (𝑓 ∘ 𝑥))
79	77, 78	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))
80	76, 79	eqtrd 2203	. . . . . . . . . . . 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))
81	73, 80	eqtrid 2215	. . . . . . . . . . 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))
82	81	eqeq2d 2182	. . . . . . . . . 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)))
83		eqcom 2172	. . . . . . . . . 10 ⊢ ((𝑦 ∘ 𝑔) = (𝑓 ∘ 𝑥) ↔ (𝑓 ∘ 𝑥) = (𝑦 ∘ 𝑔))
84	82, 83	bitrdi 195	. . . . . . . . 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)))
85	72, 84	bitr4d 190	. . . . . . . 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)))
86		f1of1 5441	. . . . . . . . . 10 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴–1-1→𝐵)
87	86	ad2antrr 485	. . . . . . . . 9 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → 𝑓:𝐴–1-1→𝐵)
88		simprl 526	. . . . . . . . 9 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → 𝑥:𝐶⟶𝐴)
89	58	adantrl 475	. . . . . . . . 9 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → (◡𝑓 ∘ (𝑦 ∘ 𝑔)):𝐶⟶𝐴)
90		cocan1 5766	. . . . . . . . 9 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝑥:𝐶⟶𝐴 ∧ (◡𝑓 ∘ (𝑦 ∘ 𝑔)):𝐶⟶𝐴) → ((𝑓 ∘ 𝑥) = (𝑓 ∘ (◡𝑓 ∘ (𝑦 ∘ 𝑔))) ↔ 𝑥 = (◡𝑓 ∘ (𝑦 ∘ 𝑔))))
91	87, 88, 89, 90	syl3anc 1233	. . . . . . . 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))))
92	24	adantr 274	. . . . . . . . 9 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → 𝑔:𝐶–onto→𝐷)
93		ffn 5347	. . . . . . . . . 10 ⊢ (𝑦:𝐷⟶𝐵 → 𝑦 Fn 𝐷)
94	93	ad2antll 488	. . . . . . . . 9 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → 𝑦 Fn 𝐷)
95	42	adantrr 476	. . . . . . . . . 10 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → ((𝑓 ∘ 𝑥) ∘ ◡𝑔):𝐷⟶𝐵)
96		ffn 5347	. . . . . . . . . 10 ⊢ (((𝑓 ∘ 𝑥) ∘ ◡𝑔):𝐷⟶𝐵 → ((𝑓 ∘ 𝑥) ∘ ◡𝑔) Fn 𝐷)
97	95, 96	syl 14	. . . . . . . . 9 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → ((𝑓 ∘ 𝑥) ∘ ◡𝑔) Fn 𝐷)
98		cocan2 5767	. . . . . . . . 9 ⊢ ((𝑔:𝐶–onto→𝐷 ∧ 𝑦 Fn 𝐷 ∧ ((𝑓 ∘ 𝑥) ∘ ◡𝑔) Fn 𝐷) → ((𝑦 ∘ 𝑔) = (((𝑓 ∘ 𝑥) ∘ ◡𝑔) ∘ 𝑔) ↔ 𝑦 = ((𝑓 ∘ 𝑥) ∘ ◡𝑔)))
99	92, 94, 97, 98	syl3anc 1233	. . . . . . . 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)))
100	85, 91, 99	3bitr3d 217	. . . . . . 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)))
101	100	ex 114	. . . . . 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))))
102	31, 47, 101	syl2ani 406	. . . . 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))))
103	16, 30, 46, 62, 102	en3d 6747	. . . 4 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → (𝐴 ↑𝑚 𝐶) ≈ (𝐵 ↑𝑚 𝐷))
104	103	exlimivv 1889	. . 3 ⊢ (∃𝑓∃𝑔(𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → (𝐴 ↑𝑚 𝐶) ≈ (𝐵 ↑𝑚 𝐷))
105	3, 104	sylbir 134	. 2 ⊢ ((∃𝑓 𝑓:𝐴–1-1-onto→𝐵 ∧ ∃𝑔 𝑔:𝐶–1-1-onto→𝐷) → (𝐴 ↑𝑚 𝐶) ≈ (𝐵 ↑𝑚 𝐷))
106	1, 2, 105	syl2anb 289	1 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 ↑𝑚 𝐶) ≈ (𝐵 ↑𝑚 𝐷))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1348  ∃wex 1485   ∈ wcel 2141  Vcvv 2730   class class class wbr 3989   I cid 4273   × cxp 4609  ^◡ccnv 4610  dom cdm 4611  ran crn 4612   ↾ cres 4613   ∘ ccom 4615   Fn wfn 5193  ⟶wf 5194  –1-1→wf1 5195  –onto→wfo 5196  –1-1-onto→wf1o 5197  (class class class)co 5853   ↑_𝑚 cmap 6626   ≈ cen 6716

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-map 6628  df-en 6719

This theorem is referenced by:  mapdom1g  6825