Theorem mapen 8124

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 7964	. 2 ⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵)
2		bren 7964	. 2 ⊢ (𝐶 ≈ 𝐷 ↔ ∃𝑔 𝑔:𝐶–1-1-onto→𝐷)
3		eeanv 2182	. . 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 6680	. . . . 5 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → (𝐴 ↑𝑚 𝐶) ∈ V)
5		ovexd 6680	. . . . 5 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → (𝐵 ↑𝑚 𝐷) ∈ V)
6		elmapi 7879	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ↑𝑚 𝐶) → 𝑥:𝐶⟶𝐴)
7		f1of 6137	. . . . . . . . . . 11 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴⟶𝐵)
8	7	adantr 481	. . . . . . . . . 10 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → 𝑓:𝐴⟶𝐵)
9		fco 6058	. . . . . . . . . 10 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥:𝐶⟶𝐴) → (𝑓 ∘ 𝑥):𝐶⟶𝐵)
10	8, 9	sylan 488	. . . . . . . . 9 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ 𝑥:𝐶⟶𝐴) → (𝑓 ∘ 𝑥):𝐶⟶𝐵)
11		f1ocnv 6149	. . . . . . . . . . . 12 ⊢ (𝑔:𝐶–1-1-onto→𝐷 → ◡𝑔:𝐷–1-1-onto→𝐶)
12	11	adantl 482	. . . . . . . . . . 11 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → ◡𝑔:𝐷–1-1-onto→𝐶)
13		f1of 6137	. . . . . . . . . . 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 481	. . . . . . . . 9 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ 𝑥:𝐶⟶𝐴) → ◡𝑔:𝐷⟶𝐶)
16		fco 6058	. . . . . . . . 9 ⊢ (((𝑓 ∘ 𝑥):𝐶⟶𝐵 ∧ ◡𝑔:𝐷⟶𝐶) → ((𝑓 ∘ 𝑥) ∘ ◡𝑔):𝐷⟶𝐵)
17	10, 15, 16	syl2anc 693	. . . . . . . 8 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ 𝑥:𝐶⟶𝐴) → ((𝑓 ∘ 𝑥) ∘ ◡𝑔):𝐷⟶𝐵)
18	17	ex 450	. . . . . . 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 6144	. . . . . . . . . 10 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴–onto→𝐵)
21	20	adantr 481	. . . . . . . . 9 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → 𝑓:𝐴–onto→𝐵)
22		forn 6118	. . . . . . . . 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 3203	. . . . . . . . 9 ⊢ 𝑓 ∈ V
25	24	rnex 7100	. . . . . . . 8 ⊢ ran 𝑓 ∈ V
26	23, 25	syl6eqelr 2710	. . . . . . 7 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → 𝐵 ∈ V)
27		f1ofo 6144	. . . . . . . . . 10 ⊢ (𝑔:𝐶–1-1-onto→𝐷 → 𝑔:𝐶–onto→𝐷)
28	27	adantl 482	. . . . . . . . 9 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → 𝑔:𝐶–onto→𝐷)
29		forn 6118	. . . . . . . . 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 3203	. . . . . . . . 9 ⊢ 𝑔 ∈ V
32	31	rnex 7100	. . . . . . . 8 ⊢ ran 𝑔 ∈ V
33	30, 32	syl6eqelr 2710	. . . . . . 7 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → 𝐷 ∈ V)
34	26, 33	elmapd 7871	. . . . . 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 249	. . . . 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 7879	. . . . . . 7 ⊢ (𝑦 ∈ (𝐵 ↑𝑚 𝐷) → 𝑦:𝐷⟶𝐵)
37		f1ocnv 6149	. . . . . . . . . . . 12 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → ◡𝑓:𝐵–1-1-onto→𝐴)
38	37	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → ◡𝑓:𝐵–1-1-onto→𝐴)
39		f1of 6137	. . . . . . . . . . 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 481	. . . . . . . . 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 6137	. . . . . . . . . . 11 ⊢ (𝑔:𝐶–1-1-onto→𝐷 → 𝑔:𝐶⟶𝐷)
44	43	adantl 482	. . . . . . . . . 10 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → 𝑔:𝐶⟶𝐷)
45		fco 6058	. . . . . . . . . 10 ⊢ ((𝑦:𝐷⟶𝐵 ∧ 𝑔:𝐶⟶𝐷) → (𝑦 ∘ 𝑔):𝐶⟶𝐵)
46	42, 44, 45	syl2anr 495	. . . . . . . . 9 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ 𝑦:𝐷⟶𝐵) → (𝑦 ∘ 𝑔):𝐶⟶𝐵)
47		fco 6058	. . . . . . . . 9 ⊢ ((◡𝑓:𝐵⟶𝐴 ∧ (𝑦 ∘ 𝑔):𝐶⟶𝐵) → (◡𝑓 ∘ (𝑦 ∘ 𝑔)):𝐶⟶𝐴)
48	41, 46, 47	syl2anc 693	. . . . . . . 8 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ 𝑦:𝐷⟶𝐵) → (◡𝑓 ∘ (𝑦 ∘ 𝑔)):𝐶⟶𝐴)
49	48	ex 450	. . . . . . 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 6141	. . . . . . . . 9 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → dom 𝑓 = 𝐴)
52	51	adantr 481	. . . . . . . 8 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → dom 𝑓 = 𝐴)
53	24	dmex 7099	. . . . . . . 8 ⊢ dom 𝑓 ∈ V
54	52, 53	syl6eqelr 2710	. . . . . . 7 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → 𝐴 ∈ V)
55		f1odm 6141	. . . . . . . . 9 ⊢ (𝑔:𝐶–1-1-onto→𝐷 → dom 𝑔 = 𝐶)
56	55	adantl 482	. . . . . . . 8 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → dom 𝑔 = 𝐶)
57	31	dmex 7099	. . . . . . . 8 ⊢ dom 𝑔 ∈ V
58	56, 57	syl6eqelr 2710	. . . . . . 7 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → 𝐶 ∈ V)
59	54, 58	elmapd 7871	. . . . . 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 249	. . . . 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 5654	. . . . . . . . . . 11 ⊢ ((𝑓 ∘ ◡𝑓) ∘ (𝑦 ∘ 𝑔)) = (𝑓 ∘ (◡𝑓 ∘ (𝑦 ∘ 𝑔)))
62		f1ococnv2 6163	. . . . . . . . . . . . . 14 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → (𝑓 ∘ ◡𝑓) = ( I ↾ 𝐵))
63	62	ad2antrr 762	. . . . . . . . . . . . 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 5283	. . . . . . . . . . . 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 752	. . . . . . . . . . . . 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 6079	. . . . . . . . . . . . 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 2656	. . . . . . . . . . 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 2670	. . . . . . . . . 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 2632	. . . . . . . . 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 5654	. . . . . . . . . . . 12 ⊢ (((𝑓 ∘ 𝑥) ∘ ◡𝑔) ∘ 𝑔) = ((𝑓 ∘ 𝑥) ∘ (◡𝑔 ∘ 𝑔))
72		f1ococnv1 6165	. . . . . . . . . . . . . . 15 ⊢ (𝑔:𝐶–1-1-onto→𝐷 → (◡𝑔 ∘ 𝑔) = ( I ↾ 𝐶))
73	72	ad2antlr 763	. . . . . . . . . . . . . 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 5284	. . . . . . . . . . . . 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 753	. . . . . . . . . . . . . 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 6078	. . . . . . . . . . . . . 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 2656	. . . . . . . . . . . 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 2668	. . . . . . . . . . 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 2632	. . . . . . . . . 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 2629	. . . . . . . . . 10 ⊢ ((𝑦 ∘ 𝑔) = (𝑓 ∘ 𝑥) ↔ (𝑓 ∘ 𝑥) = (𝑦 ∘ 𝑔))
82	80, 81	syl6bb 276	. . . . . . . . 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 271	. . . . . . . 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 6136	. . . . . . . . . 10 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴–1-1→𝐵)
85	84	ad2antrr 762	. . . . . . . . 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 794	. . . . . . . . 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 752	. . . . . . . . 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 6546	. . . . . . . . 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 1326	. . . . . . . 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 481	. . . . . . . . 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 6045	. . . . . . . . . 10 ⊢ (𝑦:𝐷⟶𝐵 → 𝑦 Fn 𝐷)
92	91	ad2antll 765	. . . . . . . . 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 753	. . . . . . . . . 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		ffn 6045	. . . . . . . . . 10 ⊢ (((𝑓 ∘ 𝑥) ∘ ◡𝑔):𝐷⟶𝐵 → ((𝑓 ∘ 𝑥) ∘ ◡𝑔) Fn 𝐷)
95	93, 94	syl 17	. . . . . . . . 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 6547	. . . . . . . . 9 ⊢ ((𝑔:𝐶–onto→𝐷 ∧ 𝑦 Fn 𝐷 ∧ ((𝑓 ∘ 𝑥) ∘ ◡𝑔) Fn 𝐷) → ((𝑦 ∘ 𝑔) = (((𝑓 ∘ 𝑥) ∘ ◡𝑔) ∘ 𝑔) ↔ 𝑦 = ((𝑓 ∘ 𝑥) ∘ ◡𝑔)))
97	90, 92, 95, 96	syl3anc 1326	. . . . . . . 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	83, 89, 97	3bitr3d 298	. . . . . . 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 450	. . . . . 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	6, 36, 99	syl2ani 688	. . . . 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	4, 5, 35, 60, 100	en3d 7992	. . . 4 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → (𝐴 ↑𝑚 𝐶) ≈ (𝐵 ↑𝑚 𝐷))
102	101	exlimivv 1860	. . 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 225	. 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 496	1 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 ↑𝑚 𝐶) ≈ (𝐵 ↑𝑚 𝐷))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  Vcvv 3200   class class class wbr 4653   I cid 5023  ^◡ccnv 5113  dom cdm 5114  ran crn 5115   ↾ cres 5116   ∘ ccom 5118   Fn wfn 5883  ⟶wf 5884  –1-1→wf1 5885  –onto→wfo 5886  –1-1-onto→wf1o 5887  (class class class)co 6650   ↑_𝑚 cmap 7857   ≈ cen 7952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859  df-en 7956

This theorem is referenced by:  mapdom1  8125  mapdom2  8131  pwen  8133  mappwen  8935  mapcdaen  9006  cfpwsdom  9406  rpnnen  14956  rexpen  14957  enrelmap  38291