Theorem mapen 9191

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 9014	. 2 ⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴⟶1-1onto𝐵)
2		bren 9014	. 2 ⊢ (𝐶 ≈ 𝐷 ↔ ∃𝑔 𝑔:𝐶⟶1-1onto𝐷)
3		exdistrv 1990	. . 3 ⊢ (∃𝑓∃𝑔(𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) ↔ (∃𝑓 𝑓:𝐴⟶1-1onto𝐵 ∧ ∃𝑔 𝑔:𝐶⟶1-1onto𝐷))
4		ovexd 7481	. . . . 5 ⊢ ((𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) → (𝐴 ↑m 𝐶) ∈ V)
5		ovexd 7481	. . . . 5 ⊢ ((𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) → (𝐵 ↑m 𝐷) ∈ V)
6		elmapi 8900	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ↑m 𝐶) → 𝑥:𝐶⟶𝐴)
7		f1of 6849	. . . . . . . . . . 11 ⊢ (𝑓:𝐴⟶1-1onto𝐵 → 𝑓:𝐴⟶𝐵)
8	7	adantr 486	. . . . . . . . . 10 ⊢ ((𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) → 𝑓:𝐴⟶𝐵)
9		fco 6759	. . . . . . . . . 10 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥:𝐶⟶𝐴) → (𝑓 ∘ 𝑥):𝐶⟶𝐵)
10	8, 9	sylan 592	. . . . . . . . 9 ⊢ (((𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) ∧ 𝑥:𝐶⟶𝐴) → (𝑓 ∘ 𝑥):𝐶⟶𝐵)
11		f1ocnv 6862	. . . . . . . . . . . 12 ⊢ (𝑔:𝐶⟶1-1onto𝐷 → ◡𝑔:𝐷⟶1-1onto𝐶)
12	11	adantl 487	. . . . . . . . . . 11 ⊢ ((𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) → ◡𝑔:𝐷⟶1-1onto𝐶)
13		f1of 6849	. . . . . . . . . . 11 ⊢ (◡𝑔:𝐷⟶1-1onto𝐶 → ◡𝑔:𝐷⟶𝐶)
14	12, 13	syl 18	. . . . . . . . . 10 ⊢ ((𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) → ◡𝑔:𝐷⟶𝐶)
15	14	adantr 486	. . . . . . . . 9 ⊢ (((𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) ∧ 𝑥:𝐶⟶𝐴) → ◡𝑔:𝐷⟶𝐶)
16	10, 15	fcod 6760	. . . . . . . 8 ⊢ (((𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) ∧ 𝑥:𝐶⟶𝐴) → ((𝑓 ∘ 𝑥) ∘ ◡𝑔):𝐷⟶𝐵)
17	16	ex 418	. . . . . . 7 ⊢ ((𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) → (𝑥:𝐶⟶𝐴 → ((𝑓 ∘ 𝑥) ∘ ◡𝑔):𝐷⟶𝐵))
18	6, 17	syl5 35	. . . . . 6 ⊢ ((𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) → (𝑥 ∈ (𝐴 ↑m 𝐶) → ((𝑓 ∘ 𝑥) ∘ ◡𝑔):𝐷⟶𝐵))
19		f1ofo 6857	. . . . . . . . . 10 ⊢ (𝑓:𝐴⟶1-1onto𝐵 → 𝑓:𝐴⟶onto𝐵)
20	19	adantr 486	. . . . . . . . 9 ⊢ ((𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) → 𝑓:𝐴⟶onto𝐵)
21		forn 6824	. . . . . . . . 9 ⊢ (𝑓:𝐴⟶onto𝐵 → ran 𝑓 = 𝐵)
22	20, 21	syl 18	. . . . . . . 8 ⊢ ((𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) → ran 𝑓 = 𝐵)
23		vex 3458	. . . . . . . . 9 ⊢ 𝑓 ∈ V
24	23	rnex 7949	. . . . . . . 8 ⊢ ran 𝑓 ∈ V
25	22, 24	eqeltrrdi 2873	. . . . . . 7 ⊢ ((𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) → 𝐵 ∈ V)
26		f1ofo 6857	. . . . . . . . . 10 ⊢ (𝑔:𝐶⟶1-1onto𝐷 → 𝑔:𝐶⟶onto𝐷)
27	26	adantl 487	. . . . . . . . 9 ⊢ ((𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) → 𝑔:𝐶⟶onto𝐷)
28		forn 6824	. . . . . . . . 9 ⊢ (𝑔:𝐶⟶onto𝐷 → ran 𝑔 = 𝐷)
29	27, 28	syl 18	. . . . . . . 8 ⊢ ((𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) → ran 𝑔 = 𝐷)
30		vex 3458	. . . . . . . . 9 ⊢ 𝑔 ∈ V
31	30	rnex 7949	. . . . . . . 8 ⊢ ran 𝑔 ∈ V
32	29, 31	eqeltrrdi 2873	. . . . . . 7 ⊢ ((𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) → 𝐷 ∈ V)
33	25, 32	elmapd 8891	. . . . . 6 ⊢ ((𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) → (((𝑓 ∘ 𝑥) ∘ ◡𝑔) ∈ (𝐵 ↑m 𝐷) ↔ ((𝑓 ∘ 𝑥) ∘ ◡𝑔):𝐷⟶𝐵))
34	18, 33	sylibrd 262	. . . . 5 ⊢ ((𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) → (𝑥 ∈ (𝐴 ↑m 𝐶) → ((𝑓 ∘ 𝑥) ∘ ◡𝑔) ∈ (𝐵 ↑m 𝐷)))
35		elmapi 8900	. . . . . . 7 ⊢ (𝑦 ∈ (𝐵 ↑m 𝐷) → 𝑦:𝐷⟶𝐵)
36		f1ocnv 6862	. . . . . . . . . . . 12 ⊢ (𝑓:𝐴⟶1-1onto𝐵 → ◡𝑓:𝐵⟶1-1onto𝐴)
37	36	adantr 486	. . . . . . . . . . 11 ⊢ ((𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) → ◡𝑓:𝐵⟶1-1onto𝐴)
38		f1of 6849	. . . . . . . . . . 11 ⊢ (◡𝑓:𝐵⟶1-1onto𝐴 → ◡𝑓:𝐵⟶𝐴)
39	37, 38	syl 18	. . . . . . . . . 10 ⊢ ((𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) → ◡𝑓:𝐵⟶𝐴)
40	39	adantr 486	. . . . . . . . 9 ⊢ (((𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) ∧ 𝑦:𝐷⟶𝐵) → ◡𝑓:𝐵⟶𝐴)
41		id 23	. . . . . . . . . 10 ⊢ (𝑦:𝐷⟶𝐵 → 𝑦:𝐷⟶𝐵)
42		f1of 6849	. . . . . . . . . . 11 ⊢ (𝑔:𝐶⟶1-1onto𝐷 → 𝑔:𝐶⟶𝐷)
43	42	adantl 487	. . . . . . . . . 10 ⊢ ((𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) → 𝑔:𝐶⟶𝐷)
44		fco 6759	. . . . . . . . . 10 ⊢ ((𝑦:𝐷⟶𝐵 ∧ 𝑔:𝐶⟶𝐷) → (𝑦 ∘ 𝑔):𝐶⟶𝐵)
45	41, 43, 44	syl2anr 609	. . . . . . . . 9 ⊢ (((𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) ∧ 𝑦:𝐷⟶𝐵) → (𝑦 ∘ 𝑔):𝐶⟶𝐵)
46	40, 45	fcod 6760	. . . . . . . 8 ⊢ (((𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) ∧ 𝑦:𝐷⟶𝐵) → (◡𝑓 ∘ (𝑦 ∘ 𝑔)):𝐶⟶𝐴)
47	46	ex 418	. . . . . . 7 ⊢ ((𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) → (𝑦:𝐷⟶𝐵 → (◡𝑓 ∘ (𝑦 ∘ 𝑔)):𝐶⟶𝐴))
48	35, 47	syl5 35	. . . . . 6 ⊢ ((𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) → (𝑦 ∈ (𝐵 ↑m 𝐷) → (◡𝑓 ∘ (𝑦 ∘ 𝑔)):𝐶⟶𝐴))
49		f1odm 6853	. . . . . . . . 9 ⊢ (𝑓:𝐴⟶1-1onto𝐵 → dom 𝑓 = 𝐴)
50	49	adantr 486	. . . . . . . 8 ⊢ ((𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) → dom 𝑓 = 𝐴)
51	23	dmex 7948	. . . . . . . 8 ⊢ dom 𝑓 ∈ V
52	50, 51	eqeltrrdi 2873	. . . . . . 7 ⊢ ((𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) → 𝐴 ∈ V)
53		f1odm 6853	. . . . . . . . 9 ⊢ (𝑔:𝐶⟶1-1onto𝐷 → dom 𝑔 = 𝐶)
54	53	adantl 487	. . . . . . . 8 ⊢ ((𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) → dom 𝑔 = 𝐶)
55	30	dmex 7948	. . . . . . . 8 ⊢ dom 𝑔 ∈ V
56	54, 55	eqeltrrdi 2873	. . . . . . 7 ⊢ ((𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) → 𝐶 ∈ V)
57	52, 56	elmapd 8891	. . . . . 6 ⊢ ((𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) → ((◡𝑓 ∘ (𝑦 ∘ 𝑔)) ∈ (𝐴 ↑m 𝐶) ↔ (◡𝑓 ∘ (𝑦 ∘ 𝑔)):𝐶⟶𝐴))
58	48, 57	sylibrd 262	. . . . 5 ⊢ ((𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) → (𝑦 ∈ (𝐵 ↑m 𝐷) → (◡𝑓 ∘ (𝑦 ∘ 𝑔)) ∈ (𝐴 ↑m 𝐶)))
59		coass 6285	. . . . . . . . . . 11 ⊢ ((𝑓 ∘ ◡𝑓) ∘ (𝑦 ∘ 𝑔)) = (𝑓 ∘ (◡𝑓 ∘ (𝑦 ∘ 𝑔)))
60		f1ococnv2 6878	. . . . . . . . . . . . . 14 ⊢ (𝑓:𝐴⟶1-1onto𝐵 → (𝑓 ∘ ◡𝑓) = ( I ↾ 𝐵))
61	60	ad2antrr 739	. . . . . . . . . . . . 13 ⊢ (((𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → (𝑓 ∘ ◡𝑓) = ( I ↾ 𝐵))
62	61	coeq1d 5846	. . . . . . . . . . . 12 ⊢ (((𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → ((𝑓 ∘ ◡𝑓) ∘ (𝑦 ∘ 𝑔)) = (( I ↾ 𝐵) ∘ (𝑦 ∘ 𝑔)))
63	45	adantrl 729	. . . . . . . . . . . . 13 ⊢ (((𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → (𝑦 ∘ 𝑔):𝐶⟶𝐵)
64		fcoi2 6782	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∘ 𝑔):𝐶⟶𝐵 → (( I ↾ 𝐵) ∘ (𝑦 ∘ 𝑔)) = (𝑦 ∘ 𝑔))
65	63, 64	syl 18	. . . . . . . . . . . 12 ⊢ (((𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → (( I ↾ 𝐵) ∘ (𝑦 ∘ 𝑔)) = (𝑦 ∘ 𝑔))
66	62, 65	eqtrd 2798	. . . . . . . . . . 11 ⊢ (((𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → ((𝑓 ∘ ◡𝑓) ∘ (𝑦 ∘ 𝑔)) = (𝑦 ∘ 𝑔))
67	59, 66	eqtr3id 2812	. . . . . . . . . 10 ⊢ (((𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → (𝑓 ∘ (◡𝑓 ∘ (𝑦 ∘ 𝑔))) = (𝑦 ∘ 𝑔))
68	67	eqeq2d 2774	. . . . . . . . 9 ⊢ (((𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → ((𝑓 ∘ 𝑥) = (𝑓 ∘ (◡𝑓 ∘ (𝑦 ∘ 𝑔))) ↔ (𝑓 ∘ 𝑥) = (𝑦 ∘ 𝑔)))
69		coass 6285	. . . . . . . . . . . 12 ⊢ (((𝑓 ∘ 𝑥) ∘ ◡𝑔) ∘ 𝑔) = ((𝑓 ∘ 𝑥) ∘ (◡𝑔 ∘ 𝑔))
70		f1ococnv1 6880	. . . . . . . . . . . . . . 15 ⊢ (𝑔:𝐶⟶1-1onto𝐷 → (◡𝑔 ∘ 𝑔) = ( I ↾ 𝐶))
71	70	ad2antlr 740	. . . . . . . . . . . . . 14 ⊢ (((𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → (◡𝑔 ∘ 𝑔) = ( I ↾ 𝐶))
72	71	coeq2d 5847	. . . . . . . . . . . . 13 ⊢ (((𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → ((𝑓 ∘ 𝑥) ∘ (◡𝑔 ∘ 𝑔)) = ((𝑓 ∘ 𝑥) ∘ ( I ↾ 𝐶)))
73	10	adantrr 730	. . . . . . . . . . . . . 14 ⊢ (((𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → (𝑓 ∘ 𝑥):𝐶⟶𝐵)
74		fcoi1 6781	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∘ 𝑥):𝐶⟶𝐵 → ((𝑓 ∘ 𝑥) ∘ ( I ↾ 𝐶)) = (𝑓 ∘ 𝑥))
75	73, 74	syl 18	. . . . . . . . . . . . 13 ⊢ (((𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → ((𝑓 ∘ 𝑥) ∘ ( I ↾ 𝐶)) = (𝑓 ∘ 𝑥))
76	72, 75	eqtrd 2798	. . . . . . . . . . . 12 ⊢ (((𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → ((𝑓 ∘ 𝑥) ∘ (◡𝑔 ∘ 𝑔)) = (𝑓 ∘ 𝑥))
77	69, 76	eqtrid 2810	. . . . . . . . . . 11 ⊢ (((𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → (((𝑓 ∘ 𝑥) ∘ ◡𝑔) ∘ 𝑔) = (𝑓 ∘ 𝑥))
78	77	eqeq2d 2774	. . . . . . . . . 10 ⊢ (((𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → ((𝑦 ∘ 𝑔) = (((𝑓 ∘ 𝑥) ∘ ◡𝑔) ∘ 𝑔) ↔ (𝑦 ∘ 𝑔) = (𝑓 ∘ 𝑥)))
79		eqcom 2770	. . . . . . . . . 10 ⊢ ((𝑦 ∘ 𝑔) = (𝑓 ∘ 𝑥) ↔ (𝑓 ∘ 𝑥) = (𝑦 ∘ 𝑔))
80	78, 79	bitrdi 290	. . . . . . . . 9 ⊢ (((𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → ((𝑦 ∘ 𝑔) = (((𝑓 ∘ 𝑥) ∘ ◡𝑔) ∘ 𝑔) ↔ (𝑓 ∘ 𝑥) = (𝑦 ∘ 𝑔)))
81	68, 80	bitr4d 285	. . . . . . . 8 ⊢ (((𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → ((𝑓 ∘ 𝑥) = (𝑓 ∘ (◡𝑓 ∘ (𝑦 ∘ 𝑔))) ↔ (𝑦 ∘ 𝑔) = (((𝑓 ∘ 𝑥) ∘ ◡𝑔) ∘ 𝑔)))
82		f1of1 6848	. . . . . . . . . 10 ⊢ (𝑓:𝐴⟶1-1onto𝐵 → 𝑓:𝐴⟶1-1𝐵)
83	82	ad2antrr 739	. . . . . . . . 9 ⊢ (((𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → 𝑓:𝐴⟶1-1𝐵)
84		simprl 783	. . . . . . . . 9 ⊢ (((𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → 𝑥:𝐶⟶𝐴)
85	46	adantrl 729	. . . . . . . . 9 ⊢ (((𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → (◡𝑓 ∘ (𝑦 ∘ 𝑔)):𝐶⟶𝐴)
86		cocan1 7325	. . . . . . . . 9 ⊢ ((𝑓:𝐴⟶1-1𝐵 ∧ 𝑥:𝐶⟶𝐴 ∧ (◡𝑓 ∘ (𝑦 ∘ 𝑔)):𝐶⟶𝐴) → ((𝑓 ∘ 𝑥) = (𝑓 ∘ (◡𝑓 ∘ (𝑦 ∘ 𝑔))) ↔ 𝑥 = (◡𝑓 ∘ (𝑦 ∘ 𝑔))))
87	83, 84, 85, 86	syl3anc 1399	. . . . . . . 8 ⊢ (((𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → ((𝑓 ∘ 𝑥) = (𝑓 ∘ (◡𝑓 ∘ (𝑦 ∘ 𝑔))) ↔ 𝑥 = (◡𝑓 ∘ (𝑦 ∘ 𝑔))))
88	27	adantr 486	. . . . . . . . 9 ⊢ (((𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → 𝑔:𝐶⟶onto𝐷)
89		ffn 6734	. . . . . . . . . 10 ⊢ (𝑦:𝐷⟶𝐵 → 𝑦 Fn 𝐷)
90	89	ad2antll 742	. . . . . . . . 9 ⊢ (((𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → 𝑦 Fn 𝐷)
91	16	adantrr 730	. . . . . . . . . 10 ⊢ (((𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → ((𝑓 ∘ 𝑥) ∘ ◡𝑔):𝐷⟶𝐵)
92	91	ffnd 6735	. . . . . . . . 9 ⊢ (((𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → ((𝑓 ∘ 𝑥) ∘ ◡𝑔) Fn 𝐷)
93		cocan2 7326	. . . . . . . . 9 ⊢ ((𝑔:𝐶⟶onto𝐷 ∧ 𝑦 Fn 𝐷 ∧ ((𝑓 ∘ 𝑥) ∘ ◡𝑔) Fn 𝐷) → ((𝑦 ∘ 𝑔) = (((𝑓 ∘ 𝑥) ∘ ◡𝑔) ∘ 𝑔) ↔ 𝑦 = ((𝑓 ∘ 𝑥) ∘ ◡𝑔)))
94	88, 90, 92, 93	syl3anc 1399	. . . . . . . 8 ⊢ (((𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → ((𝑦 ∘ 𝑔) = (((𝑓 ∘ 𝑥) ∘ ◡𝑔) ∘ 𝑔) ↔ 𝑦 = ((𝑓 ∘ 𝑥) ∘ ◡𝑔)))
95	81, 87, 94	3bitr3d 312	. . . . . . 7 ⊢ (((𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → (𝑥 = (◡𝑓 ∘ (𝑦 ∘ 𝑔)) ↔ 𝑦 = ((𝑓 ∘ 𝑥) ∘ ◡𝑔)))
96	95	ex 418	. . . . . 6 ⊢ ((𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) → ((𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵) → (𝑥 = (◡𝑓 ∘ (𝑦 ∘ 𝑔)) ↔ 𝑦 = ((𝑓 ∘ 𝑥) ∘ ◡𝑔))))
97	6, 35, 96	syl2ani 619	. . . . 5 ⊢ ((𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) → ((𝑥 ∈ (𝐴 ↑m 𝐶) ∧ 𝑦 ∈ (𝐵 ↑m 𝐷)) → (𝑥 = (◡𝑓 ∘ (𝑦 ∘ 𝑔)) ↔ 𝑦 = ((𝑓 ∘ 𝑥) ∘ ◡𝑔))))
98	4, 5, 34, 58, 97	en3d 9047	. . . 4 ⊢ ((𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) → (𝐴 ↑m 𝐶) ≈ (𝐵 ↑m 𝐷))
99	98	exlimivv 1967	. . 3 ⊢ (∃𝑓∃𝑔(𝑓:𝐴⟶1-1onto𝐵 ∧ 𝑔:𝐶⟶1-1onto𝐷) → (𝐴 ↑m 𝐶) ≈ (𝐵 ↑m 𝐷))
100	3, 99	sylbir 238	. 2 ⊢ ((∃𝑓 𝑓:𝐴⟶1-1onto𝐵 ∧ ∃𝑔 𝑔:𝐶⟶1-1onto𝐷) → (𝐴 ↑m 𝐶) ≈ (𝐵 ↑m 𝐷))
101	1, 2, 100	syl2anb 610	1 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 ↑m 𝐶) ≈ (𝐵 ↑m 𝐷))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147  Vcvv 3454   class class class wbr 5106   I cid 5550  ^◡ccnv 5655  dom cdm 5656  ran crn 5657   ↾ cres 5658   ∘ ccom 5660   Fn wfn 6556  ⟶wf 6557  ⟶1-1wf1 6558  ⟶ontowfo 6559  ⟶1-1ontowf1o 6560  (class class class)co 7446   ↑_m cmap 8878   ≈ cen 9001

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 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

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 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7449  df-oprab 7450  df-mpo 7451  df-1st 8028  df-2nd 8029  df-map 8880  df-en 9005

This theorem is used by:  mapdom1  9192  mapdom2  9198  pwen  9200  mappwen  10221  mapdjuen  10289  cfpwsdom  10699  rpnnen  16427  rexpen  16428  enrelmap  45200