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Theorem xpmapenlem 9114

Description: Lemma for xpmapen 9115. (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 7423	. 2 ⊢ ((𝐴 × 𝐵) ↑_m 𝐶) ∈ V
2		ovex 7423	. . 3 ⊢ (𝐴 ↑_m 𝐶) ∈ V
3		ovex 7423	. . 3 ⊢ (𝐵 ↑_m 𝐶) ∈ V
4	2, 3	xpex 7732	. 2 ⊢ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶)) ∈ V
5		xpmapen.1	. . . . . . . . 9 ⊢ 𝐴 ∈ V
6		xpmapen.2	. . . . . . . . 9 ⊢ 𝐵 ∈ V
7	5, 6	xpex 7732	. . . . . . . 8 ⊢ (𝐴 × 𝐵) ∈ V
8		xpmapen.3	. . . . . . . 8 ⊢ 𝐶 ∈ V
9	7, 8	elmap 8847	. . . . . . 7 ⊢ (𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ↔ 𝑥:𝐶⟶(𝐴 × 𝐵))
10		ffvelcdm 7056	. . . . . . 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 8003	. . . . . 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 7089	. . . 4 ⊢ (𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) → 𝐷:𝐶⟶𝐴)
16	5, 8	elmap 8847	. . . 4 ⊢ (𝐷 ∈ (𝐴 ↑_m 𝐶) ↔ 𝐷:𝐶⟶𝐴)
17	15, 16	sylibr 234	. . 3 ⊢ (𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) → 𝐷 ∈ (𝐴 ↑_m 𝐶))
18		xp2nd 8004	. . . . . 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 7089	. . . 4 ⊢ (𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) → 𝑅:𝐶⟶𝐵)
22	6, 8	elmap 8847	. . . 4 ⊢ (𝑅 ∈ (𝐵 ↑_m 𝐶) ↔ 𝑅:𝐶⟶𝐵)
23	21, 22	sylibr 234	. . 3 ⊢ (𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) → 𝑅 ∈ (𝐵 ↑_m 𝐶))
24	17, 23	opelxpd 5680	. 2 ⊢ (𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) → ⟨𝐷, 𝑅⟩ ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶)))
25		xp1st 8003	. . . . . . 7 ⊢ (𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶)) → (1^st ‘𝑦) ∈ (𝐴 ↑_m 𝐶))
26	5, 8	elmap 8847	. . . . . . 7 ⊢ ((1^st ‘𝑦) ∈ (𝐴 ↑_m 𝐶) ↔ (1^st ‘𝑦):𝐶⟶𝐴)
27	25, 26	sylib 218	. . . . . 6 ⊢ (𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶)) → (1^st ‘𝑦):𝐶⟶𝐴)
28	27	ffvelcdmda 7059	. . . . 5 ⊢ ((𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶)) ∧ 𝑧 ∈ 𝐶) → ((1^st ‘𝑦)‘𝑧) ∈ 𝐴)
29		xp2nd 8004	. . . . . . 7 ⊢ (𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶)) → (2^nd ‘𝑦) ∈ (𝐵 ↑_m 𝐶))
30	6, 8	elmap 8847	. . . . . . 7 ⊢ ((2^nd ‘𝑦) ∈ (𝐵 ↑_m 𝐶) ↔ (2^nd ‘𝑦):𝐶⟶𝐵)
31	29, 30	sylib 218	. . . . . 6 ⊢ (𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶)) → (2^nd ‘𝑦):𝐶⟶𝐵)
32	31	ffvelcdmda 7059	. . . . 5 ⊢ ((𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶)) ∧ 𝑧 ∈ 𝐶) → ((2^nd ‘𝑦)‘𝑧) ∈ 𝐵)
33	28, 32	opelxpd 5680	. . . 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 7089	. . 3 ⊢ (𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶)) → 𝑆:𝐶⟶(𝐴 × 𝐵))
36	7, 8	elmap 8847	. . 3 ⊢ (𝑆 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ↔ 𝑆:𝐶⟶(𝐴 × 𝐵))
37	35, 36	sylibr 234	. 2 ⊢ (𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶)) → 𝑆 ∈ ((𝐴 × 𝐵) ↑_m 𝐶))
38		1st2nd2 8010	. . . . 5 ⊢ (𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶)) → 𝑦 = ⟨(1^st ‘𝑦), (2^nd ‘𝑦)⟩)
39	38	ad2antlr 727	. . . 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 6932	. . . . . . 7 ⊢ (𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶)) → (1^st ‘𝑦) = (𝑧 ∈ 𝐶 ↦ ((1^st ‘𝑦)‘𝑧)))
41	40	ad2antlr 727	. . . . . 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 768	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑥 = 𝑆) ∧ 𝑧 ∈ 𝐶) → 𝑥 = 𝑆)
43	42	fveq1d 6863	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑥 = 𝑆) ∧ 𝑧 ∈ 𝐶) → (𝑥‘𝑧) = (𝑆‘𝑧))
44		opex 5427	. . . . . . . . . . . . 13 ⊢ ⟨((1^st ‘𝑦)‘𝑧), ((2^nd ‘𝑦)‘𝑧)⟩ ∈ V
45	34	fvmpt2 6982	. . . . . . . . . . . . 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 691	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝐶 → (𝑆‘𝑧) = ⟨((1^st ‘𝑦)‘𝑧), ((2^nd ‘𝑦)‘𝑧)⟩)
47	46	adantl 481	. . . . . . . . . . 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 2765	. . . . . . . . . 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 6865	. . . . . . . . 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 6874	. . . . . . . . . 10 ⊢ ((1^st ‘𝑦)‘𝑧) ∈ V
51		fvex 6874	. . . . . . . . . 10 ⊢ ((2^nd ‘𝑦)‘𝑧) ∈ V
52	50, 51	op1st 7979	. . . . . . . . 9 ⊢ (1^st ‘⟨((1^st ‘𝑦)‘𝑧), ((2^nd ‘𝑦)‘𝑧)⟩) = ((1^st ‘𝑦)‘𝑧)
53	49, 52	eqtrdi 2781	. . . . . . . 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 5203	. . . . . . 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 2777	. . . . . 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 2768	. . . . 5 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑥 = 𝑆) → (1^st ‘𝑦) = 𝐷)
57	31	feqmptd 6932	. . . . . . 7 ⊢ (𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶)) → (2^nd ‘𝑦) = (𝑧 ∈ 𝐶 ↦ ((2^nd ‘𝑦)‘𝑧)))
58	57	ad2antlr 727	. . . . . 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 6865	. . . . . . . . 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 7980	. . . . . . . . 9 ⊢ (2^nd ‘⟨((1^st ‘𝑦)‘𝑧), ((2^nd ‘𝑦)‘𝑧)⟩) = ((2^nd ‘𝑦)‘𝑧)
61	59, 60	eqtrdi 2781	. . . . . . . 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 5203	. . . . . . 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 2777	. . . . . 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 2768	. . . . 5 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑥 = 𝑆) → (2^nd ‘𝑦) = 𝑅)
65	56, 64	opeq12d 4848	. . . 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 2765	. . 3 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑥 = 𝑆) → 𝑦 = ⟨𝐷, 𝑅⟩)
67		simpll 766	. . . . . 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 218	. . . . 5 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → 𝑥:𝐶⟶(𝐴 × 𝐵))
69	68	feqmptd 6932	. . . 4 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → 𝑥 = (𝑧 ∈ 𝐶 ↦ (𝑥‘𝑧)))
70		simpr 484	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → 𝑦 = ⟨𝐷, 𝑅⟩)
71	70	fveq2d 6865	. . . . . . . . . . 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 726	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → 𝐷 ∈ (𝐴 ↑_m 𝐶))
73	23	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → 𝑅 ∈ (𝐵 ↑_m 𝐶))
74		op1stg 7983	. . . . . . . . . . . 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 2765	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → (1^st ‘𝑦) = 𝐷)
77	76	fveq1d 6863	. . . . . . . . 9 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → ((1^st ‘𝑦)‘𝑧) = (𝐷‘𝑧))
78		fvex 6874	. . . . . . . . . 10 ⊢ (1^st ‘(𝑥‘𝑧)) ∈ V
79	14	fvmpt2 6982	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝐶 ∧ (1^st ‘(𝑥‘𝑧)) ∈ V) → (𝐷‘𝑧) = (1^st ‘(𝑥‘𝑧)))
80	78, 79	mpan2 691	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐶 → (𝐷‘𝑧) = (1^st ‘(𝑥‘𝑧)))
81	77, 80	sylan9eq 2785	. . . . . . . 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 6865	. . . . . . . . . . 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 7984	. . . . . . . . . . . 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 2765	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → (2^nd ‘𝑦) = 𝑅)
86	85	fveq1d 6863	. . . . . . . . 9 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → ((2^nd ‘𝑦)‘𝑧) = (𝑅‘𝑧))
87		fvex 6874	. . . . . . . . . 10 ⊢ (2^nd ‘(𝑥‘𝑧)) ∈ V
88	20	fvmpt2 6982	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝐶 ∧ (2^nd ‘(𝑥‘𝑧)) ∈ V) → (𝑅‘𝑧) = (2^nd ‘(𝑥‘𝑧)))
89	87, 88	mpan2 691	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐶 → (𝑅‘𝑧) = (2^nd ‘(𝑥‘𝑧)))
90	86, 89	sylan9eq 2785	. . . . . . . 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 4848	. . . . . . 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 7059	. . . . . . . 8 ⊢ ((((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) ∧ 𝑧 ∈ 𝐶) → (𝑥‘𝑧) ∈ (𝐴 × 𝐵))
93		1st2nd2 8010	. . . . . . . 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 2768	. . . . . 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 5203	. . . . 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 2777	. . . 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 2768	. . 3 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → 𝑥 = 𝑆)
99	66, 98	impbida 800	. 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 8965	1 ⊢ ((𝐴 × 𝐵) ↑_m 𝐶) ≈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))

Colors of variables: wff setvar class

Syntax hints: ∧ wa 395 = wceq 1540 ∈ wcel 2109 Vcvv 3450 ⟨cop 4598 class class class wbr 5110 ↦ cmpt 5191 × cxp 5639 ⟶wf 6510 ‘cfv 6514 (class class class)co 7390 1^st c1st 7969 2^nd c2nd 7970 ↑_m cmap 8802 ≈ cen 8918

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5254 ax-nul 5264 ax-pow 5323 ax-pr 5390 ax-un 7714

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-rab 3409 df-v 3452 df-sbc 3757 df-csb 3866 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-nul 4300 df-if 4492 df-pw 4568 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-br 5111 df-opab 5173 df-mpt 5192 df-id 5536 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-ima 5654 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-f1 6519 df-fo 6520 df-f1o 6521 df-fv 6522 df-ov 7393 df-oprab 7394 df-mpo 7395 df-1st 7971 df-2nd 7972 df-map 8804 df-en 8922

This theorem is referenced by: xpmapen 9115

W3C validator