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Theorem mapen 8665

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 8501	. 2 ⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵)
2		bren 8501	. 2 ⊢ (𝐶 ≈ 𝐷 ↔ ∃𝑔 𝑔:𝐶–1-1-onto→𝐷)
3		exdistrv 1956	. . 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		ovexd 7170	. . . . 5 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → (𝐴 ↑_m 𝐶) ∈ V)
5		ovexd 7170	. . . . 5 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → (𝐵 ↑_m 𝐷) ∈ V)
6		elmapi 8411	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ↑_m 𝐶) → 𝑥:𝐶⟶𝐴)
7		f1of 6590	. . . . . . . . . . 11 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴⟶𝐵)
8	7	adantr 484	. . . . . . . . . 10 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → 𝑓:𝐴⟶𝐵)
9		fco 6505	. . . . . . . . . 10 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥:𝐶⟶𝐴) → (𝑓 ∘ 𝑥):𝐶⟶𝐵)
10	8, 9	sylan 583	. . . . . . . . 9 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ 𝑥:𝐶⟶𝐴) → (𝑓 ∘ 𝑥):𝐶⟶𝐵)
11		f1ocnv 6602	. . . . . . . . . . . 12 ⊢ (𝑔:𝐶–1-1-onto→𝐷 → ^◡𝑔:𝐷–1-1-onto→𝐶)
12	11	adantl 485	. . . . . . . . . . 11 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → ^◡𝑔:𝐷–1-1-onto→𝐶)
13		f1of 6590	. . . . . . . . . . 11 ⊢ (^◡𝑔:𝐷–1-1-onto→𝐶 → ^◡𝑔:𝐷⟶𝐶)
14	12, 13	syl 17	. . . . . . . . . 10 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → ^◡𝑔:𝐷⟶𝐶)
15	14	adantr 484	. . . . . . . . 9 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ 𝑥:𝐶⟶𝐴) → ^◡𝑔:𝐷⟶𝐶)
16		fco 6505	. . . . . . . . 9 ⊢ (((𝑓 ∘ 𝑥):𝐶⟶𝐵 ∧ ^◡𝑔:𝐷⟶𝐶) → ((𝑓 ∘ 𝑥) ∘ ^◡𝑔):𝐷⟶𝐵)
17	10, 15, 16	syl2anc 587	. . . . . . . 8 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ 𝑥:𝐶⟶𝐴) → ((𝑓 ∘ 𝑥) ∘ ^◡𝑔):𝐷⟶𝐵)
18	17	ex 416	. . . . . . 7 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → (𝑥:𝐶⟶𝐴 → ((𝑓 ∘ 𝑥) ∘ ^◡𝑔):𝐷⟶𝐵))
19	6, 18	syl5 34	. . . . . 6 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → (𝑥 ∈ (𝐴 ↑_m 𝐶) → ((𝑓 ∘ 𝑥) ∘ ^◡𝑔):𝐷⟶𝐵))
20		f1ofo 6597	. . . . . . . . . 10 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴–onto→𝐵)
21	20	adantr 484	. . . . . . . . 9 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → 𝑓:𝐴–onto→𝐵)
22		forn 6568	. . . . . . . . 9 ⊢ (𝑓:𝐴–onto→𝐵 → ran 𝑓 = 𝐵)
23	21, 22	syl 17	. . . . . . . 8 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → ran 𝑓 = 𝐵)
24		vex 3444	. . . . . . . . 9 ⊢ 𝑓 ∈ V
25	24	rnex 7599	. . . . . . . 8 ⊢ ran 𝑓 ∈ V
26	23, 25	eqeltrrdi 2899	. . . . . . 7 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → 𝐵 ∈ V)
27		f1ofo 6597	. . . . . . . . . 10 ⊢ (𝑔:𝐶–1-1-onto→𝐷 → 𝑔:𝐶–onto→𝐷)
28	27	adantl 485	. . . . . . . . 9 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → 𝑔:𝐶–onto→𝐷)
29		forn 6568	. . . . . . . . 9 ⊢ (𝑔:𝐶–onto→𝐷 → ran 𝑔 = 𝐷)
30	28, 29	syl 17	. . . . . . . 8 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → ran 𝑔 = 𝐷)
31		vex 3444	. . . . . . . . 9 ⊢ 𝑔 ∈ V
32	31	rnex 7599	. . . . . . . 8 ⊢ ran 𝑔 ∈ V
33	30, 32	eqeltrrdi 2899	. . . . . . 7 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → 𝐷 ∈ V)
34	26, 33	elmapd 8403	. . . . . 6 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → (((𝑓 ∘ 𝑥) ∘ ^◡𝑔) ∈ (𝐵 ↑_m 𝐷) ↔ ((𝑓 ∘ 𝑥) ∘ ^◡𝑔):𝐷⟶𝐵))
35	19, 34	sylibrd 262	. . . . 5 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → (𝑥 ∈ (𝐴 ↑_m 𝐶) → ((𝑓 ∘ 𝑥) ∘ ^◡𝑔) ∈ (𝐵 ↑_m 𝐷)))
36		elmapi 8411	. . . . . . 7 ⊢ (𝑦 ∈ (𝐵 ↑_m 𝐷) → 𝑦:𝐷⟶𝐵)
37		f1ocnv 6602	. . . . . . . . . . . 12 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → ^◡𝑓:𝐵–1-1-onto→𝐴)
38	37	adantr 484	. . . . . . . . . . 11 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → ^◡𝑓:𝐵–1-1-onto→𝐴)
39		f1of 6590	. . . . . . . . . . 11 ⊢ (^◡𝑓:𝐵–1-1-onto→𝐴 → ^◡𝑓:𝐵⟶𝐴)
40	38, 39	syl 17	. . . . . . . . . 10 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → ^◡𝑓:𝐵⟶𝐴)
41	40	adantr 484	. . . . . . . . 9 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ 𝑦:𝐷⟶𝐵) → ^◡𝑓:𝐵⟶𝐴)
42		id 22	. . . . . . . . . 10 ⊢ (𝑦:𝐷⟶𝐵 → 𝑦:𝐷⟶𝐵)
43		f1of 6590	. . . . . . . . . . 11 ⊢ (𝑔:𝐶–1-1-onto→𝐷 → 𝑔:𝐶⟶𝐷)
44	43	adantl 485	. . . . . . . . . 10 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → 𝑔:𝐶⟶𝐷)
45		fco 6505	. . . . . . . . . 10 ⊢ ((𝑦:𝐷⟶𝐵 ∧ 𝑔:𝐶⟶𝐷) → (𝑦 ∘ 𝑔):𝐶⟶𝐵)
46	42, 44, 45	syl2anr 599	. . . . . . . . 9 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ 𝑦:𝐷⟶𝐵) → (𝑦 ∘ 𝑔):𝐶⟶𝐵)
47		fco 6505	. . . . . . . . 9 ⊢ ((^◡𝑓:𝐵⟶𝐴 ∧ (𝑦 ∘ 𝑔):𝐶⟶𝐵) → (^◡𝑓 ∘ (𝑦 ∘ 𝑔)):𝐶⟶𝐴)
48	41, 46, 47	syl2anc 587	. . . . . . . 8 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ 𝑦:𝐷⟶𝐵) → (^◡𝑓 ∘ (𝑦 ∘ 𝑔)):𝐶⟶𝐴)
49	48	ex 416	. . . . . . 7 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → (𝑦:𝐷⟶𝐵 → (^◡𝑓 ∘ (𝑦 ∘ 𝑔)):𝐶⟶𝐴))
50	36, 49	syl5 34	. . . . . 6 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → (𝑦 ∈ (𝐵 ↑_m 𝐷) → (^◡𝑓 ∘ (𝑦 ∘ 𝑔)):𝐶⟶𝐴))
51		f1odm 6594	. . . . . . . . 9 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → dom 𝑓 = 𝐴)
52	51	adantr 484	. . . . . . . 8 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → dom 𝑓 = 𝐴)
53	24	dmex 7598	. . . . . . . 8 ⊢ dom 𝑓 ∈ V
54	52, 53	eqeltrrdi 2899	. . . . . . 7 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → 𝐴 ∈ V)
55		f1odm 6594	. . . . . . . . 9 ⊢ (𝑔:𝐶–1-1-onto→𝐷 → dom 𝑔 = 𝐶)
56	55	adantl 485	. . . . . . . 8 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → dom 𝑔 = 𝐶)
57	31	dmex 7598	. . . . . . . 8 ⊢ dom 𝑔 ∈ V
58	56, 57	eqeltrrdi 2899	. . . . . . 7 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → 𝐶 ∈ V)
59	54, 58	elmapd 8403	. . . . . 6 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → ((^◡𝑓 ∘ (𝑦 ∘ 𝑔)) ∈ (𝐴 ↑_m 𝐶) ↔ (^◡𝑓 ∘ (𝑦 ∘ 𝑔)):𝐶⟶𝐴))
60	50, 59	sylibrd 262	. . . . 5 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → (𝑦 ∈ (𝐵 ↑_m 𝐷) → (^◡𝑓 ∘ (𝑦 ∘ 𝑔)) ∈ (𝐴 ↑_m 𝐶)))
61		coass 6085	. . . . . . . . . . 11 ⊢ ((𝑓 ∘ ^◡𝑓) ∘ (𝑦 ∘ 𝑔)) = (𝑓 ∘ (^◡𝑓 ∘ (𝑦 ∘ 𝑔)))
62		f1ococnv2 6616	. . . . . . . . . . . . . 14 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → (𝑓 ∘ ^◡𝑓) = ( I ↾ 𝐵))
63	62	ad2antrr 725	. . . . . . . . . . . . 13 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → (𝑓 ∘ ^◡𝑓) = ( I ↾ 𝐵))
64	63	coeq1d 5696	. . . . . . . . . . . 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)))
65	46	adantrl 715	. . . . . . . . . . . . 13 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → (𝑦 ∘ 𝑔):𝐶⟶𝐵)
66		fcoi2 6527	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∘ 𝑔):𝐶⟶𝐵 → (( I ↾ 𝐵) ∘ (𝑦 ∘ 𝑔)) = (𝑦 ∘ 𝑔))
67	65, 66	syl 17	. . . . . . . . . . . 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))
68	64, 67	eqtrd 2833	. . . . . . . . . . 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))
69	61, 68	syl5eqr 2847	. . . . . . . . . 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))
70	69	eqeq2d 2809	. . . . . . . . 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)))
71		coass 6085	. . . . . . . . . . . 12 ⊢ (((𝑓 ∘ 𝑥) ∘ ^◡𝑔) ∘ 𝑔) = ((𝑓 ∘ 𝑥) ∘ (^◡𝑔 ∘ 𝑔))
72		f1ococnv1 6618	. . . . . . . . . . . . . . 15 ⊢ (𝑔:𝐶–1-1-onto→𝐷 → (^◡𝑔 ∘ 𝑔) = ( I ↾ 𝐶))
73	72	ad2antlr 726	. . . . . . . . . . . . . 14 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → (^◡𝑔 ∘ 𝑔) = ( I ↾ 𝐶))
74	73	coeq2d 5697	. . . . . . . . . . . . 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)))
75	10	adantrr 716	. . . . . . . . . . . . . 14 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → (𝑓 ∘ 𝑥):𝐶⟶𝐵)
76		fcoi1 6526	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∘ 𝑥):𝐶⟶𝐵 → ((𝑓 ∘ 𝑥) ∘ ( I ↾ 𝐶)) = (𝑓 ∘ 𝑥))
77	75, 76	syl 17	. . . . . . . . . . . . 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))
78	74, 77	eqtrd 2833	. . . . . . . . . . . 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))
79	71, 78	syl5eq 2845	. . . . . . . . . . 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))
80	79	eqeq2d 2809	. . . . . . . . . 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)))
81		eqcom 2805	. . . . . . . . . 10 ⊢ ((𝑦 ∘ 𝑔) = (𝑓 ∘ 𝑥) ↔ (𝑓 ∘ 𝑥) = (𝑦 ∘ 𝑔))
82	80, 81	syl6bb 290	. . . . . . . . 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)))
83	70, 82	bitr4d 285	. . . . . . . 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)))
84		f1of1 6589	. . . . . . . . . 10 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴–1-1→𝐵)
85	84	ad2antrr 725	. . . . . . . . 9 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → 𝑓:𝐴–1-1→𝐵)
86		simprl 770	. . . . . . . . 9 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → 𝑥:𝐶⟶𝐴)
87	48	adantrl 715	. . . . . . . . 9 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → (^◡𝑓 ∘ (𝑦 ∘ 𝑔)):𝐶⟶𝐴)
88		cocan1 7025	. . . . . . . . 9 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝑥:𝐶⟶𝐴 ∧ (^◡𝑓 ∘ (𝑦 ∘ 𝑔)):𝐶⟶𝐴) → ((𝑓 ∘ 𝑥) = (𝑓 ∘ (^◡𝑓 ∘ (𝑦 ∘ 𝑔))) ↔ 𝑥 = (^◡𝑓 ∘ (𝑦 ∘ 𝑔))))
89	85, 86, 87, 88	syl3anc 1368	. . . . . . . 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))))
90	28	adantr 484	. . . . . . . . 9 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → 𝑔:𝐶–onto→𝐷)
91		ffn 6487	. . . . . . . . . 10 ⊢ (𝑦:𝐷⟶𝐵 → 𝑦 Fn 𝐷)
92	91	ad2antll 728	. . . . . . . . 9 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → 𝑦 Fn 𝐷)
93	17	adantrr 716	. . . . . . . . . 10 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → ((𝑓 ∘ 𝑥) ∘ ^◡𝑔):𝐷⟶𝐵)
94	93	ffnd 6488	. . . . . . . . 9 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → ((𝑓 ∘ 𝑥) ∘ ^◡𝑔) Fn 𝐷)
95		cocan2 7026	. . . . . . . . 9 ⊢ ((𝑔:𝐶–onto→𝐷 ∧ 𝑦 Fn 𝐷 ∧ ((𝑓 ∘ 𝑥) ∘ ^◡𝑔) Fn 𝐷) → ((𝑦 ∘ 𝑔) = (((𝑓 ∘ 𝑥) ∘ ^◡𝑔) ∘ 𝑔) ↔ 𝑦 = ((𝑓 ∘ 𝑥) ∘ ^◡𝑔)))
96	90, 92, 94, 95	syl3anc 1368	. . . . . . . 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)))
97	83, 89, 96	3bitr3d 312	. . . . . . 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)))
98	97	ex 416	. . . . . 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))))
99	6, 36, 98	syl2ani 609	. . . . 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))))
100	4, 5, 35, 60, 99	en3d 8529	. . . 4 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → (𝐴 ↑_m 𝐶) ≈ (𝐵 ↑_m 𝐷))
101	100	exlimivv 1933	. . 3 ⊢ (∃𝑓∃𝑔(𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → (𝐴 ↑_m 𝐶) ≈ (𝐵 ↑_m 𝐷))
102	3, 101	sylbir 238	. 2 ⊢ ((∃𝑓 𝑓:𝐴–1-1-onto→𝐵 ∧ ∃𝑔 𝑔:𝐶–1-1-onto→𝐷) → (𝐴 ↑_m 𝐶) ≈ (𝐵 ↑_m 𝐷))
103	1, 2, 102	syl2anb 600	1 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 ↑_m 𝐶) ≈ (𝐵 ↑_m 𝐷))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∃wex 1781 ∈ wcel 2111 Vcvv 3441 class class class wbr 5030 I cid 5424 ^◡ccnv 5518 dom cdm 5519 ran crn 5520 ↾ cres 5521 ∘ ccom 5523 Fn wfn 6319 ⟶wf 6320 –1-1→wf1 6321 –onto→wfo 6322 –1-1-onto→wf1o 6323 (class class class)co 7135 ↑_m cmap 8389 ≈ cen 8489

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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-map 8391 df-en 8493

This theorem is referenced by: mapdom1 8666 mapdom2 8672 pwen 8674 mappwen 9523 mapdjuen 9591 cfpwsdom 9995 rpnnen 15572 rexpen 15573 enrelmap 40698

W3C validator