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Theorem xpmapenlem 9088

Description: Lemma for xpmapen 9089. (Contributed by NM, 1-May-2004.) (Revised by Mario Carneiro, 16-Nov-2014.)

**Hypotheses**
Ref	Expression
xpmapen.1	⊢ 𝐴 ∈ V
xpmapen.2	⊢ 𝐵 ∈ V
xpmapen.3	⊢ 𝐶 ∈ V
xpmapenlem.4	⊢ 𝐷 = (𝑧 ∈ 𝐶 ↦ (1^st ‘(𝑥‘𝑧)))
xpmapenlem.5	⊢ 𝑅 = (𝑧 ∈ 𝐶 ↦ (2^nd ‘(𝑥‘𝑧)))
xpmapenlem.6	⊢ 𝑆 = (𝑧 ∈ 𝐶 ↦ ⟨((1^st ‘𝑦)‘𝑧), ((2^nd ‘𝑦)‘𝑧)⟩)

**Assertion**
Ref	Expression
xpmapenlem	⊢ ((𝐴 × 𝐵) ↑_m 𝐶) ≈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))

Distinct variable groups: 𝑥,𝑦,𝑧,𝐴 𝑥,𝐵,𝑦,𝑧 𝑥,𝐶,𝑦,𝑧 𝑦,𝐷,𝑧 𝑦,𝑅,𝑧 𝑥,𝑆,𝑧 Allowed substitution hints: 𝐷(𝑥) 𝑅(𝑥) 𝑆(𝑦)

**Proof of Theorem xpmapenlem**
Step	Hyp	Ref	Expression
1		ovex 7390	. 2 ⊢ ((𝐴 × 𝐵) ↑_m 𝐶) ∈ V
2		ovex 7390	. . 3 ⊢ (𝐴 ↑_m 𝐶) ∈ V
3		ovex 7390	. . 3 ⊢ (𝐵 ↑_m 𝐶) ∈ V
4	2, 3	xpex 7687	. 2 ⊢ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶)) ∈ V
5		xpmapen.1	. . . . . . . . 9 ⊢ 𝐴 ∈ V
6		xpmapen.2	. . . . . . . . 9 ⊢ 𝐵 ∈ V
7	5, 6	xpex 7687	. . . . . . . 8 ⊢ (𝐴 × 𝐵) ∈ V
8		xpmapen.3	. . . . . . . 8 ⊢ 𝐶 ∈ V
9	7, 8	elmap 8809	. . . . . . 7 ⊢ (𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ↔ 𝑥:𝐶⟶(𝐴 × 𝐵))
10		ffvelcdm 7032	. . . . . . 7 ⊢ ((𝑥:𝐶⟶(𝐴 × 𝐵) ∧ 𝑧 ∈ 𝐶) → (𝑥‘𝑧) ∈ (𝐴 × 𝐵))
11	9, 10	sylanb 581	. . . . . 6 ⊢ ((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑧 ∈ 𝐶) → (𝑥‘𝑧) ∈ (𝐴 × 𝐵))
12		xp1st 7953	. . . . . 6 ⊢ ((𝑥‘𝑧) ∈ (𝐴 × 𝐵) → (1^st ‘(𝑥‘𝑧)) ∈ 𝐴)
13	11, 12	syl 17	. . . . 5 ⊢ ((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑧 ∈ 𝐶) → (1^st ‘(𝑥‘𝑧)) ∈ 𝐴)
14		xpmapenlem.4	. . . . 5 ⊢ 𝐷 = (𝑧 ∈ 𝐶 ↦ (1^st ‘(𝑥‘𝑧)))
15	13, 14	fmptd 7062	. . . 4 ⊢ (𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) → 𝐷:𝐶⟶𝐴)
16	5, 8	elmap 8809	. . . 4 ⊢ (𝐷 ∈ (𝐴 ↑_m 𝐶) ↔ 𝐷:𝐶⟶𝐴)
17	15, 16	sylibr 233	. . 3 ⊢ (𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) → 𝐷 ∈ (𝐴 ↑_m 𝐶))
18		xp2nd 7954	. . . . . 6 ⊢ ((𝑥‘𝑧) ∈ (𝐴 × 𝐵) → (2^nd ‘(𝑥‘𝑧)) ∈ 𝐵)
19	11, 18	syl 17	. . . . 5 ⊢ ((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑧 ∈ 𝐶) → (2^nd ‘(𝑥‘𝑧)) ∈ 𝐵)
20		xpmapenlem.5	. . . . 5 ⊢ 𝑅 = (𝑧 ∈ 𝐶 ↦ (2^nd ‘(𝑥‘𝑧)))
21	19, 20	fmptd 7062	. . . 4 ⊢ (𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) → 𝑅:𝐶⟶𝐵)
22	6, 8	elmap 8809	. . . 4 ⊢ (𝑅 ∈ (𝐵 ↑_m 𝐶) ↔ 𝑅:𝐶⟶𝐵)
23	21, 22	sylibr 233	. . 3 ⊢ (𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) → 𝑅 ∈ (𝐵 ↑_m 𝐶))
24	17, 23	opelxpd 5671	. 2 ⊢ (𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) → ⟨𝐷, 𝑅⟩ ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶)))
25		xp1st 7953	. . . . . . 7 ⊢ (𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶)) → (1^st ‘𝑦) ∈ (𝐴 ↑_m 𝐶))
26	5, 8	elmap 8809	. . . . . . 7 ⊢ ((1^st ‘𝑦) ∈ (𝐴 ↑_m 𝐶) ↔ (1^st ‘𝑦):𝐶⟶𝐴)
27	25, 26	sylib 217	. . . . . 6 ⊢ (𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶)) → (1^st ‘𝑦):𝐶⟶𝐴)
28	27	ffvelcdmda 7035	. . . . 5 ⊢ ((𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶)) ∧ 𝑧 ∈ 𝐶) → ((1^st ‘𝑦)‘𝑧) ∈ 𝐴)
29		xp2nd 7954	. . . . . . 7 ⊢ (𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶)) → (2^nd ‘𝑦) ∈ (𝐵 ↑_m 𝐶))
30	6, 8	elmap 8809	. . . . . . 7 ⊢ ((2^nd ‘𝑦) ∈ (𝐵 ↑_m 𝐶) ↔ (2^nd ‘𝑦):𝐶⟶𝐵)
31	29, 30	sylib 217	. . . . . 6 ⊢ (𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶)) → (2^nd ‘𝑦):𝐶⟶𝐵)
32	31	ffvelcdmda 7035	. . . . 5 ⊢ ((𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶)) ∧ 𝑧 ∈ 𝐶) → ((2^nd ‘𝑦)‘𝑧) ∈ 𝐵)
33	28, 32	opelxpd 5671	. . . 4 ⊢ ((𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶)) ∧ 𝑧 ∈ 𝐶) → ⟨((1^st ‘𝑦)‘𝑧), ((2^nd ‘𝑦)‘𝑧)⟩ ∈ (𝐴 × 𝐵))
34		xpmapenlem.6	. . . 4 ⊢ 𝑆 = (𝑧 ∈ 𝐶 ↦ ⟨((1^st ‘𝑦)‘𝑧), ((2^nd ‘𝑦)‘𝑧)⟩)
35	33, 34	fmptd 7062	. . 3 ⊢ (𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶)) → 𝑆:𝐶⟶(𝐴 × 𝐵))
36	7, 8	elmap 8809	. . 3 ⊢ (𝑆 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ↔ 𝑆:𝐶⟶(𝐴 × 𝐵))
37	35, 36	sylibr 233	. 2 ⊢ (𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶)) → 𝑆 ∈ ((𝐴 × 𝐵) ↑_m 𝐶))
38		1st2nd2 7960	. . . . 5 ⊢ (𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶)) → 𝑦 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩)
39	38	ad2antlr 725	. . . 4 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑥 = 𝑆) → 𝑦 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩)
40	27	feqmptd 6910	. . . . . . 7 ⊢ (𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶)) → (1^st ‘𝑦) = (𝑧 ∈ 𝐶 ↦ ((1^st ‘𝑦)‘𝑧)))
41	40	ad2antlr 725	. . . . . 6 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑥 = 𝑆) → (1^st ‘𝑦) = (𝑧 ∈ 𝐶 ↦ ((1^st ‘𝑦)‘𝑧)))
42		simplr 767	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑥 = 𝑆) ∧ 𝑧 ∈ 𝐶) → 𝑥 = 𝑆)
43	42	fveq1d 6844	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑥 = 𝑆) ∧ 𝑧 ∈ 𝐶) → (𝑥‘𝑧) = (𝑆‘𝑧))
44		opex 5421	. . . . . . . . . . . . 13 ⊢ ⟨((1^st ‘𝑦)‘𝑧), ((2^nd ‘𝑦)‘𝑧)⟩ ∈ V
45	34	fvmpt2 6959	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ 𝐶 ∧ ⟨((1^st ‘𝑦)‘𝑧), ((2^nd ‘𝑦)‘𝑧)⟩ ∈ V) → (𝑆‘𝑧) = ⟨((1^st ‘𝑦)‘𝑧), ((2^nd ‘𝑦)‘𝑧)⟩)
46	44, 45	mpan2 689	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝐶 → (𝑆‘𝑧) = ⟨((1^st ‘𝑦)‘𝑧), ((2^nd ‘𝑦)‘𝑧)⟩)
47	46	adantl 482	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑥 = 𝑆) ∧ 𝑧 ∈ 𝐶) → (𝑆‘𝑧) = ⟨((1^st ‘𝑦)‘𝑧), ((2^nd ‘𝑦)‘𝑧)⟩)
48	43, 47	eqtrd 2776	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑥 = 𝑆) ∧ 𝑧 ∈ 𝐶) → (𝑥‘𝑧) = ⟨((1^st ‘𝑦)‘𝑧), ((2^nd ‘𝑦)‘𝑧)⟩)
49	48	fveq2d 6846	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑥 = 𝑆) ∧ 𝑧 ∈ 𝐶) → (1^st ‘(𝑥‘𝑧)) = (1^st ‘⟨((1^st ‘𝑦)‘𝑧), ((2^nd ‘𝑦)‘𝑧)⟩))
50		fvex 6855	. . . . . . . . . 10 ⊢ ((1^st ‘𝑦)‘𝑧) ∈ V
51		fvex 6855	. . . . . . . . . 10 ⊢ ((2^nd ‘𝑦)‘𝑧) ∈ V
52	50, 51	op1st 7929	. . . . . . . . 9 ⊢ (1^st ‘⟨((1^st ‘𝑦)‘𝑧), ((2^nd ‘𝑦)‘𝑧)⟩) = ((1^st ‘𝑦)‘𝑧)
53	49, 52	eqtrdi 2792	. . . . . . . 8 ⊢ ((((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑥 = 𝑆) ∧ 𝑧 ∈ 𝐶) → (1^st ‘(𝑥‘𝑧)) = ((1^st ‘𝑦)‘𝑧))
54	53	mpteq2dva 5205	. . . . . . 7 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑥 = 𝑆) → (𝑧 ∈ 𝐶 ↦ (1^st ‘(𝑥‘𝑧))) = (𝑧 ∈ 𝐶 ↦ ((1^st ‘𝑦)‘𝑧)))
55	14, 54	eqtrid 2788	. . . . . 6 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑥 = 𝑆) → 𝐷 = (𝑧 ∈ 𝐶 ↦ ((1^st ‘𝑦)‘𝑧)))
56	41, 55	eqtr4d 2779	. . . . 5 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑥 = 𝑆) → (1^st ‘𝑦) = 𝐷)
57	31	feqmptd 6910	. . . . . . 7 ⊢ (𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶)) → (2^nd ‘𝑦) = (𝑧 ∈ 𝐶 ↦ ((2^nd ‘𝑦)‘𝑧)))
58	57	ad2antlr 725	. . . . . 6 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑥 = 𝑆) → (2^nd ‘𝑦) = (𝑧 ∈ 𝐶 ↦ ((2^nd ‘𝑦)‘𝑧)))
59	48	fveq2d 6846	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑥 = 𝑆) ∧ 𝑧 ∈ 𝐶) → (2^nd ‘(𝑥‘𝑧)) = (2^nd ‘⟨((1^st ‘𝑦)‘𝑧), ((2^nd ‘𝑦)‘𝑧)⟩))
60	50, 51	op2nd 7930	. . . . . . . . 9 ⊢ (2^nd ‘⟨((1^st ‘𝑦)‘𝑧), ((2^nd ‘𝑦)‘𝑧)⟩) = ((2^nd ‘𝑦)‘𝑧)
61	59, 60	eqtrdi 2792	. . . . . . . 8 ⊢ ((((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑥 = 𝑆) ∧ 𝑧 ∈ 𝐶) → (2^nd ‘(𝑥‘𝑧)) = ((2^nd ‘𝑦)‘𝑧))
62	61	mpteq2dva 5205	. . . . . . 7 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑥 = 𝑆) → (𝑧 ∈ 𝐶 ↦ (2^nd ‘(𝑥‘𝑧))) = (𝑧 ∈ 𝐶 ↦ ((2^nd ‘𝑦)‘𝑧)))
63	20, 62	eqtrid 2788	. . . . . 6 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑥 = 𝑆) → 𝑅 = (𝑧 ∈ 𝐶 ↦ ((2^nd ‘𝑦)‘𝑧)))
64	58, 63	eqtr4d 2779	. . . . 5 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑥 = 𝑆) → (2^nd ‘𝑦) = 𝑅)
65	56, 64	opeq12d 4838	. . . 4 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑥 = 𝑆) → ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩ = ⟨𝐷, 𝑅⟩)
66	39, 65	eqtrd 2776	. . 3 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑥 = 𝑆) → 𝑦 = ⟨𝐷, 𝑅⟩)
67		simpll 765	. . . . . 6 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → 𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶))
68	67, 9	sylib 217	. . . . 5 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → 𝑥:𝐶⟶(𝐴 × 𝐵))
69	68	feqmptd 6910	. . . 4 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → 𝑥 = (𝑧 ∈ 𝐶 ↦ (𝑥‘𝑧)))
70		simpr 485	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → 𝑦 = ⟨𝐷, 𝑅⟩)
71	70	fveq2d 6846	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → (1^st ‘𝑦) = (1^st ‘⟨𝐷, 𝑅⟩))
72	17	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → 𝐷 ∈ (𝐴 ↑_m 𝐶))
73	23	ad2antrr 724	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → 𝑅 ∈ (𝐵 ↑_m 𝐶))
74		op1stg 7933	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (𝐴 ↑_m 𝐶) ∧ 𝑅 ∈ (𝐵 ↑_m 𝐶)) → (1^st ‘⟨𝐷, 𝑅⟩) = 𝐷)
75	72, 73, 74	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → (1^st ‘⟨𝐷, 𝑅⟩) = 𝐷)
76	71, 75	eqtrd 2776	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → (1^st ‘𝑦) = 𝐷)
77	76	fveq1d 6844	. . . . . . . . 9 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → ((1^st ‘𝑦)‘𝑧) = (𝐷‘𝑧))
78		fvex 6855	. . . . . . . . . 10 ⊢ (1^st ‘(𝑥‘𝑧)) ∈ V
79	14	fvmpt2 6959	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝐶 ∧ (1^st ‘(𝑥‘𝑧)) ∈ V) → (𝐷‘𝑧) = (1^st ‘(𝑥‘𝑧)))
80	78, 79	mpan2 689	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐶 → (𝐷‘𝑧) = (1^st ‘(𝑥‘𝑧)))
81	77, 80	sylan9eq 2796	. . . . . . . 8 ⊢ ((((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) ∧ 𝑧 ∈ 𝐶) → ((1^st ‘𝑦)‘𝑧) = (1^st ‘(𝑥‘𝑧)))
82	70	fveq2d 6846	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → (2^nd ‘𝑦) = (2^nd ‘⟨𝐷, 𝑅⟩))
83		op2ndg 7934	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (𝐴 ↑_m 𝐶) ∧ 𝑅 ∈ (𝐵 ↑_m 𝐶)) → (2^nd ‘⟨𝐷, 𝑅⟩) = 𝑅)
84	72, 73, 83	syl2anc 584	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → (2^nd ‘⟨𝐷, 𝑅⟩) = 𝑅)
85	82, 84	eqtrd 2776	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → (2^nd ‘𝑦) = 𝑅)
86	85	fveq1d 6844	. . . . . . . . 9 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → ((2^nd ‘𝑦)‘𝑧) = (𝑅‘𝑧))
87		fvex 6855	. . . . . . . . . 10 ⊢ (2^nd ‘(𝑥‘𝑧)) ∈ V
88	20	fvmpt2 6959	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝐶 ∧ (2^nd ‘(𝑥‘𝑧)) ∈ V) → (𝑅‘𝑧) = (2^nd ‘(𝑥‘𝑧)))
89	87, 88	mpan2 689	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐶 → (𝑅‘𝑧) = (2^nd ‘(𝑥‘𝑧)))
90	86, 89	sylan9eq 2796	. . . . . . . 8 ⊢ ((((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) ∧ 𝑧 ∈ 𝐶) → ((2^nd ‘𝑦)‘𝑧) = (2^nd ‘(𝑥‘𝑧)))
91	81, 90	opeq12d 4838	. . . . . . 7 ⊢ ((((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) ∧ 𝑧 ∈ 𝐶) → ⟨((1^st ‘𝑦)‘𝑧), ((2^nd ‘𝑦)‘𝑧)⟩ = ⟨(1^st ‘(𝑥‘𝑧)), (2^nd ‘(𝑥‘𝑧))⟩)
92	68	ffvelcdmda 7035	. . . . . . . 8 ⊢ ((((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) ∧ 𝑧 ∈ 𝐶) → (𝑥‘𝑧) ∈ (𝐴 × 𝐵))
93		1st2nd2 7960	. . . . . . . 8 ⊢ ((𝑥‘𝑧) ∈ (𝐴 × 𝐵) → (𝑥‘𝑧) = ⟨(1^st ‘(𝑥‘𝑧)), (2^nd ‘(𝑥‘𝑧))⟩)
94	92, 93	syl 17	. . . . . . 7 ⊢ ((((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) ∧ 𝑧 ∈ 𝐶) → (𝑥‘𝑧) = ⟨(1^st ‘(𝑥‘𝑧)), (2^nd ‘(𝑥‘𝑧))⟩)
95	91, 94	eqtr4d 2779	. . . . . 6 ⊢ ((((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) ∧ 𝑧 ∈ 𝐶) → ⟨((1^st ‘𝑦)‘𝑧), ((2^nd ‘𝑦)‘𝑧)⟩ = (𝑥‘𝑧))
96	95	mpteq2dva 5205	. . . . 5 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → (𝑧 ∈ 𝐶 ↦ ⟨((1^st ‘𝑦)‘𝑧), ((2^nd ‘𝑦)‘𝑧)⟩) = (𝑧 ∈ 𝐶 ↦ (𝑥‘𝑧)))
97	34, 96	eqtrid 2788	. . . 4 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → 𝑆 = (𝑧 ∈ 𝐶 ↦ (𝑥‘𝑧)))
98	69, 97	eqtr4d 2779	. . 3 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → 𝑥 = 𝑆)
99	66, 98	impbida 799	. 2 ⊢ ((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) → (𝑥 = 𝑆 ↔ 𝑦 = ⟨𝐷, 𝑅⟩))
100	1, 4, 24, 37, 99	en3i 8931	1 ⊢ ((𝐴 × 𝐵) ↑_m 𝐶) ≈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))

Colors of variables: wff setvar class

Syntax hints: ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3445 ⟨cop 4592 class class class wbr 5105 ↦ cmpt 5188 × cxp 5631 ⟶wf 6492 ‘cfv 6496 (class class class)co 7357 1^st c1st 7919 2^nd c2nd 7920 ↑_m cmap 8765 ≈ cen 8880

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 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672

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 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-ov 7360 df-oprab 7361 df-mpo 7362 df-1st 7921 df-2nd 7922 df-map 8767 df-en 8884

This theorem is referenced by: xpmapen 9089

W3C validator