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Theorem mapen 9081

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 8889	. 2 ⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵)
2		bren 8889	. 2 ⊢ (𝐶 ≈ 𝐷 ↔ ∃𝑔 𝑔:𝐶–1-1-onto→𝐷)
3		exdistrv 1959	. . 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 7388	. . . . 5 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → (𝐴 ↑_m 𝐶) ∈ V)
5		ovexd 7388	. . . . 5 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → (𝐵 ↑_m 𝐷) ∈ V)
6		elmapi 8783	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ↑_m 𝐶) → 𝑥:𝐶⟶𝐴)
7		f1of 6781	. . . . . . . . . . 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 6689	. . . . . . . . . 10 ⊢ ((𝑓:𝐴⟶𝐵 ∧ 𝑥:𝐶⟶𝐴) → (𝑓 ∘ 𝑥):𝐶⟶𝐵)
10	8, 9	sylan 580	. . . . . . . . 9 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ 𝑥:𝐶⟶𝐴) → (𝑓 ∘ 𝑥):𝐶⟶𝐵)
11		f1ocnv 6793	. . . . . . . . . . . 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 6781	. . . . . . . . . . 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	10, 15	fcod 6691	. . . . . . . 8 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ 𝑥:𝐶⟶𝐴) → ((𝑓 ∘ 𝑥) ∘ ^◡𝑔):𝐷⟶𝐵)
17	16	ex 413	. . . . . . 7 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → (𝑥:𝐶⟶𝐴 → ((𝑓 ∘ 𝑥) ∘ ^◡𝑔):𝐷⟶𝐵))
18	6, 17	syl5 34	. . . . . 6 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → (𝑥 ∈ (𝐴 ↑_m 𝐶) → ((𝑓 ∘ 𝑥) ∘ ^◡𝑔):𝐷⟶𝐵))
19		f1ofo 6788	. . . . . . . . . 10 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴–onto→𝐵)
20	19	adantr 481	. . . . . . . . 9 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → 𝑓:𝐴–onto→𝐵)
21		forn 6756	. . . . . . . . 9 ⊢ (𝑓:𝐴–onto→𝐵 → ran 𝑓 = 𝐵)
22	20, 21	syl 17	. . . . . . . 8 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → ran 𝑓 = 𝐵)
23		vex 3447	. . . . . . . . 9 ⊢ 𝑓 ∈ V
24	23	rnex 7845	. . . . . . . 8 ⊢ ran 𝑓 ∈ V
25	22, 24	eqeltrrdi 2847	. . . . . . 7 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → 𝐵 ∈ V)
26		f1ofo 6788	. . . . . . . . . 10 ⊢ (𝑔:𝐶–1-1-onto→𝐷 → 𝑔:𝐶–onto→𝐷)
27	26	adantl 482	. . . . . . . . 9 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → 𝑔:𝐶–onto→𝐷)
28		forn 6756	. . . . . . . . 9 ⊢ (𝑔:𝐶–onto→𝐷 → ran 𝑔 = 𝐷)
29	27, 28	syl 17	. . . . . . . 8 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → ran 𝑔 = 𝐷)
30		vex 3447	. . . . . . . . 9 ⊢ 𝑔 ∈ V
31	30	rnex 7845	. . . . . . . 8 ⊢ ran 𝑔 ∈ V
32	29, 31	eqeltrrdi 2847	. . . . . . 7 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → 𝐷 ∈ V)
33	25, 32	elmapd 8775	. . . . . 6 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → (((𝑓 ∘ 𝑥) ∘ ^◡𝑔) ∈ (𝐵 ↑_m 𝐷) ↔ ((𝑓 ∘ 𝑥) ∘ ^◡𝑔):𝐷⟶𝐵))
34	18, 33	sylibrd 258	. . . . 5 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → (𝑥 ∈ (𝐴 ↑_m 𝐶) → ((𝑓 ∘ 𝑥) ∘ ^◡𝑔) ∈ (𝐵 ↑_m 𝐷)))
35		elmapi 8783	. . . . . . 7 ⊢ (𝑦 ∈ (𝐵 ↑_m 𝐷) → 𝑦:𝐷⟶𝐵)
36		f1ocnv 6793	. . . . . . . . . . . 12 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → ^◡𝑓:𝐵–1-1-onto→𝐴)
37	36	adantr 481	. . . . . . . . . . 11 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → ^◡𝑓:𝐵–1-1-onto→𝐴)
38		f1of 6781	. . . . . . . . . . 11 ⊢ (^◡𝑓:𝐵–1-1-onto→𝐴 → ^◡𝑓:𝐵⟶𝐴)
39	37, 38	syl 17	. . . . . . . . . 10 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → ^◡𝑓:𝐵⟶𝐴)
40	39	adantr 481	. . . . . . . . 9 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ 𝑦:𝐷⟶𝐵) → ^◡𝑓:𝐵⟶𝐴)
41		id 22	. . . . . . . . . 10 ⊢ (𝑦:𝐷⟶𝐵 → 𝑦:𝐷⟶𝐵)
42		f1of 6781	. . . . . . . . . . 11 ⊢ (𝑔:𝐶–1-1-onto→𝐷 → 𝑔:𝐶⟶𝐷)
43	42	adantl 482	. . . . . . . . . 10 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → 𝑔:𝐶⟶𝐷)
44		fco 6689	. . . . . . . . . 10 ⊢ ((𝑦:𝐷⟶𝐵 ∧ 𝑔:𝐶⟶𝐷) → (𝑦 ∘ 𝑔):𝐶⟶𝐵)
45	41, 43, 44	syl2anr 597	. . . . . . . . 9 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ 𝑦:𝐷⟶𝐵) → (𝑦 ∘ 𝑔):𝐶⟶𝐵)
46	40, 45	fcod 6691	. . . . . . . 8 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ 𝑦:𝐷⟶𝐵) → (^◡𝑓 ∘ (𝑦 ∘ 𝑔)):𝐶⟶𝐴)
47	46	ex 413	. . . . . . 7 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → (𝑦:𝐷⟶𝐵 → (^◡𝑓 ∘ (𝑦 ∘ 𝑔)):𝐶⟶𝐴))
48	35, 47	syl5 34	. . . . . 6 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → (𝑦 ∈ (𝐵 ↑_m 𝐷) → (^◡𝑓 ∘ (𝑦 ∘ 𝑔)):𝐶⟶𝐴))
49		f1odm 6785	. . . . . . . . 9 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → dom 𝑓 = 𝐴)
50	49	adantr 481	. . . . . . . 8 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → dom 𝑓 = 𝐴)
51	23	dmex 7844	. . . . . . . 8 ⊢ dom 𝑓 ∈ V
52	50, 51	eqeltrrdi 2847	. . . . . . 7 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → 𝐴 ∈ V)
53		f1odm 6785	. . . . . . . . 9 ⊢ (𝑔:𝐶–1-1-onto→𝐷 → dom 𝑔 = 𝐶)
54	53	adantl 482	. . . . . . . 8 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → dom 𝑔 = 𝐶)
55	30	dmex 7844	. . . . . . . 8 ⊢ dom 𝑔 ∈ V
56	54, 55	eqeltrrdi 2847	. . . . . . 7 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → 𝐶 ∈ V)
57	52, 56	elmapd 8775	. . . . . 6 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → ((^◡𝑓 ∘ (𝑦 ∘ 𝑔)) ∈ (𝐴 ↑_m 𝐶) ↔ (^◡𝑓 ∘ (𝑦 ∘ 𝑔)):𝐶⟶𝐴))
58	48, 57	sylibrd 258	. . . . 5 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → (𝑦 ∈ (𝐵 ↑_m 𝐷) → (^◡𝑓 ∘ (𝑦 ∘ 𝑔)) ∈ (𝐴 ↑_m 𝐶)))
59		coass 6215	. . . . . . . . . . 11 ⊢ ((𝑓 ∘ ^◡𝑓) ∘ (𝑦 ∘ 𝑔)) = (𝑓 ∘ (^◡𝑓 ∘ (𝑦 ∘ 𝑔)))
60		f1ococnv2 6808	. . . . . . . . . . . . . 14 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → (𝑓 ∘ ^◡𝑓) = ( I ↾ 𝐵))
61	60	ad2antrr 724	. . . . . . . . . . . . 13 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → (𝑓 ∘ ^◡𝑓) = ( I ↾ 𝐵))
62	61	coeq1d 5815	. . . . . . . . . . . 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)))
63	45	adantrl 714	. . . . . . . . . . . . 13 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → (𝑦 ∘ 𝑔):𝐶⟶𝐵)
64		fcoi2 6714	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∘ 𝑔):𝐶⟶𝐵 → (( I ↾ 𝐵) ∘ (𝑦 ∘ 𝑔)) = (𝑦 ∘ 𝑔))
65	63, 64	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))
66	62, 65	eqtrd 2776	. . . . . . . . . . 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))
67	59, 66	eqtr3id 2790	. . . . . . . . . 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))
68	67	eqeq2d 2747	. . . . . . . . 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)))
69		coass 6215	. . . . . . . . . . . 12 ⊢ (((𝑓 ∘ 𝑥) ∘ ^◡𝑔) ∘ 𝑔) = ((𝑓 ∘ 𝑥) ∘ (^◡𝑔 ∘ 𝑔))
70		f1ococnv1 6810	. . . . . . . . . . . . . . 15 ⊢ (𝑔:𝐶–1-1-onto→𝐷 → (^◡𝑔 ∘ 𝑔) = ( I ↾ 𝐶))
71	70	ad2antlr 725	. . . . . . . . . . . . . 14 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → (^◡𝑔 ∘ 𝑔) = ( I ↾ 𝐶))
72	71	coeq2d 5816	. . . . . . . . . . . . 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)))
73	10	adantrr 715	. . . . . . . . . . . . . 14 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → (𝑓 ∘ 𝑥):𝐶⟶𝐵)
74		fcoi1 6713	. . . . . . . . . . . . . 14 ⊢ ((𝑓 ∘ 𝑥):𝐶⟶𝐵 → ((𝑓 ∘ 𝑥) ∘ ( I ↾ 𝐶)) = (𝑓 ∘ 𝑥))
75	73, 74	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))
76	72, 75	eqtrd 2776	. . . . . . . . . . . 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))
77	69, 76	eqtrid 2788	. . . . . . . . . . 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))
78	77	eqeq2d 2747	. . . . . . . . . 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)))
79		eqcom 2743	. . . . . . . . . 10 ⊢ ((𝑦 ∘ 𝑔) = (𝑓 ∘ 𝑥) ↔ (𝑓 ∘ 𝑥) = (𝑦 ∘ 𝑔))
80	78, 79	bitrdi 286	. . . . . . . . 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)))
81	68, 80	bitr4d 281	. . . . . . . 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)))
82		f1of1 6780	. . . . . . . . . 10 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓:𝐴–1-1→𝐵)
83	82	ad2antrr 724	. . . . . . . . 9 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → 𝑓:𝐴–1-1→𝐵)
84		simprl 769	. . . . . . . . 9 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → 𝑥:𝐶⟶𝐴)
85	46	adantrl 714	. . . . . . . . 9 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → (^◡𝑓 ∘ (𝑦 ∘ 𝑔)):𝐶⟶𝐴)
86		cocan1 7233	. . . . . . . . 9 ⊢ ((𝑓:𝐴–1-1→𝐵 ∧ 𝑥:𝐶⟶𝐴 ∧ (^◡𝑓 ∘ (𝑦 ∘ 𝑔)):𝐶⟶𝐴) → ((𝑓 ∘ 𝑥) = (𝑓 ∘ (^◡𝑓 ∘ (𝑦 ∘ 𝑔))) ↔ 𝑥 = (^◡𝑓 ∘ (𝑦 ∘ 𝑔))))
87	83, 84, 85, 86	syl3anc 1371	. . . . . . . 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))))
88	27	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)
89		ffn 6665	. . . . . . . . . 10 ⊢ (𝑦:𝐷⟶𝐵 → 𝑦 Fn 𝐷)
90	89	ad2antll 727	. . . . . . . . 9 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → 𝑦 Fn 𝐷)
91	16	adantrr 715	. . . . . . . . . 10 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → ((𝑓 ∘ 𝑥) ∘ ^◡𝑔):𝐷⟶𝐵)
92	91	ffnd 6666	. . . . . . . . 9 ⊢ (((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) ∧ (𝑥:𝐶⟶𝐴 ∧ 𝑦:𝐷⟶𝐵)) → ((𝑓 ∘ 𝑥) ∘ ^◡𝑔) Fn 𝐷)
93		cocan2 7234	. . . . . . . . 9 ⊢ ((𝑔:𝐶–onto→𝐷 ∧ 𝑦 Fn 𝐷 ∧ ((𝑓 ∘ 𝑥) ∘ ^◡𝑔) Fn 𝐷) → ((𝑦 ∘ 𝑔) = (((𝑓 ∘ 𝑥) ∘ ^◡𝑔) ∘ 𝑔) ↔ 𝑦 = ((𝑓 ∘ 𝑥) ∘ ^◡𝑔)))
94	88, 90, 92, 93	syl3anc 1371	. . . . . . . 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)))
95	81, 87, 94	3bitr3d 308	. . . . . . 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)))
96	95	ex 413	. . . . . 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))))
97	6, 35, 96	syl2ani 607	. . . . 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))))
98	4, 5, 34, 58, 97	en3d 8925	. . . 4 ⊢ ((𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → (𝐴 ↑_m 𝐶) ≈ (𝐵 ↑_m 𝐷))
99	98	exlimivv 1935	. . 3 ⊢ (∃𝑓∃𝑔(𝑓:𝐴–1-1-onto→𝐵 ∧ 𝑔:𝐶–1-1-onto→𝐷) → (𝐴 ↑_m 𝐶) ≈ (𝐵 ↑_m 𝐷))
100	3, 99	sylbir 234	. 2 ⊢ ((∃𝑓 𝑓:𝐴–1-1-onto→𝐵 ∧ ∃𝑔 𝑔:𝐶–1-1-onto→𝐷) → (𝐴 ↑_m 𝐶) ≈ (𝐵 ↑_m 𝐷))
101	1, 2, 100	syl2anb 598	1 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 ↑_m 𝐶) ≈ (𝐵 ↑_m 𝐷))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∃wex 1781 ∈ wcel 2106 Vcvv 3443 class class class wbr 5103 I cid 5528 ^◡ccnv 5630 dom cdm 5631 ran crn 5632 ↾ cres 5633 ∘ ccom 5635 Fn wfn 6488 ⟶wf 6489 –1-1→wf1 6490 –onto→wfo 6491 –1-1-onto→wf1o 6492 (class class class)co 7353 ↑_m cmap 8761 ≈ cen 8876

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-sbc 3738 df-csb 3854 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-iun 4954 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-ima 5644 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-ov 7356 df-oprab 7357 df-mpo 7358 df-1st 7917 df-2nd 7918 df-map 8763 df-en 8880

This theorem is referenced by: mapdom1 9082 mapdom2 9088 pwen 9090 mappwen 10044 mapdjuen 10112 cfpwsdom 10516 rpnnen 16101 rexpen 16102 enrelmap 42211

W3C validator