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Theorem xpmapenlem 8686

Description: Lemma for xpmapen 8687. (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 7191	. 2 ⊢ ((𝐴 × 𝐵) ↑_m 𝐶) ∈ V
2		ovex 7191	. . 3 ⊢ (𝐴 ↑_m 𝐶) ∈ V
3		ovex 7191	. . 3 ⊢ (𝐵 ↑_m 𝐶) ∈ V
4	2, 3	xpex 7478	. 2 ⊢ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶)) ∈ V
5		xpmapen.1	. . . . . . . . 9 ⊢ 𝐴 ∈ V
6		xpmapen.2	. . . . . . . . 9 ⊢ 𝐵 ∈ V
7	5, 6	xpex 7478	. . . . . . . 8 ⊢ (𝐴 × 𝐵) ∈ V
8		xpmapen.3	. . . . . . . 8 ⊢ 𝐶 ∈ V
9	7, 8	elmap 8437	. . . . . . 7 ⊢ (𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ↔ 𝑥:𝐶⟶(𝐴 × 𝐵))
10		ffvelrn 6851	. . . . . . 7 ⊢ ((𝑥:𝐶⟶(𝐴 × 𝐵) ∧ 𝑧 ∈ 𝐶) → (𝑥‘𝑧) ∈ (𝐴 × 𝐵))
11	9, 10	sylanb 583	. . . . . 6 ⊢ ((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑧 ∈ 𝐶) → (𝑥‘𝑧) ∈ (𝐴 × 𝐵))
12		xp1st 7723	. . . . . 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 6880	. . . 4 ⊢ (𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) → 𝐷:𝐶⟶𝐴)
16	5, 8	elmap 8437	. . . 4 ⊢ (𝐷 ∈ (𝐴 ↑_m 𝐶) ↔ 𝐷:𝐶⟶𝐴)
17	15, 16	sylibr 236	. . 3 ⊢ (𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) → 𝐷 ∈ (𝐴 ↑_m 𝐶))
18		xp2nd 7724	. . . . . 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 6880	. . . 4 ⊢ (𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) → 𝑅:𝐶⟶𝐵)
22	6, 8	elmap 8437	. . . 4 ⊢ (𝑅 ∈ (𝐵 ↑_m 𝐶) ↔ 𝑅:𝐶⟶𝐵)
23	21, 22	sylibr 236	. . 3 ⊢ (𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) → 𝑅 ∈ (𝐵 ↑_m 𝐶))
24	17, 23	opelxpd 5595	. 2 ⊢ (𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) → ⟨𝐷, 𝑅⟩ ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶)))
25		xp1st 7723	. . . . . . 7 ⊢ (𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶)) → (1^st ‘𝑦) ∈ (𝐴 ↑_m 𝐶))
26	5, 8	elmap 8437	. . . . . . 7 ⊢ ((1^st ‘𝑦) ∈ (𝐴 ↑_m 𝐶) ↔ (1^st ‘𝑦):𝐶⟶𝐴)
27	25, 26	sylib 220	. . . . . 6 ⊢ (𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶)) → (1^st ‘𝑦):𝐶⟶𝐴)
28	27	ffvelrnda 6853	. . . . 5 ⊢ ((𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶)) ∧ 𝑧 ∈ 𝐶) → ((1^st ‘𝑦)‘𝑧) ∈ 𝐴)
29		xp2nd 7724	. . . . . . 7 ⊢ (𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶)) → (2^nd ‘𝑦) ∈ (𝐵 ↑_m 𝐶))
30	6, 8	elmap 8437	. . . . . . 7 ⊢ ((2^nd ‘𝑦) ∈ (𝐵 ↑_m 𝐶) ↔ (2^nd ‘𝑦):𝐶⟶𝐵)
31	29, 30	sylib 220	. . . . . 6 ⊢ (𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶)) → (2^nd ‘𝑦):𝐶⟶𝐵)
32	31	ffvelrnda 6853	. . . . 5 ⊢ ((𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶)) ∧ 𝑧 ∈ 𝐶) → ((2^nd ‘𝑦)‘𝑧) ∈ 𝐵)
33	28, 32	opelxpd 5595	. . . 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 6880	. . 3 ⊢ (𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶)) → 𝑆:𝐶⟶(𝐴 × 𝐵))
36	7, 8	elmap 8437	. . 3 ⊢ (𝑆 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ↔ 𝑆:𝐶⟶(𝐴 × 𝐵))
37	35, 36	sylibr 236	. 2 ⊢ (𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶)) → 𝑆 ∈ ((𝐴 × 𝐵) ↑_m 𝐶))
38		1st2nd2 7730	. . . . 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 6735	. . . . . . 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 6674	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑥 = 𝑆) ∧ 𝑧 ∈ 𝐶) → (𝑥‘𝑧) = (𝑆‘𝑧))
44		opex 5358	. . . . . . . . . . . . 13 ⊢ ⟨((1^st ‘𝑦)‘𝑧), ((2^nd ‘𝑦)‘𝑧)⟩ ∈ V
45	34	fvmpt2 6781	. . . . . . . . . . . . 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 484	. . . . . . . . . . 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 2858	. . . . . . . . . 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 6676	. . . . . . . . 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 6685	. . . . . . . . . 10 ⊢ ((1^st ‘𝑦)‘𝑧) ∈ V
51		fvex 6685	. . . . . . . . . 10 ⊢ ((2^nd ‘𝑦)‘𝑧) ∈ V
52	50, 51	op1st 7699	. . . . . . . . 9 ⊢ (1^st ‘⟨((1^st ‘𝑦)‘𝑧), ((2^nd ‘𝑦)‘𝑧)⟩) = ((1^st ‘𝑦)‘𝑧)
53	49, 52	syl6eq 2874	. . . . . . . 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 5163	. . . . . . 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	syl5eq 2870	. . . . . 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 2861	. . . . 5 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑥 = 𝑆) → (1^st ‘𝑦) = 𝐷)
57	31	feqmptd 6735	. . . . . . 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 6676	. . . . . . . . 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 7700	. . . . . . . . 9 ⊢ (2^nd ‘⟨((1^st ‘𝑦)‘𝑧), ((2^nd ‘𝑦)‘𝑧)⟩) = ((2^nd ‘𝑦)‘𝑧)
61	59, 60	syl6eq 2874	. . . . . . . 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 5163	. . . . . . 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	syl5eq 2870	. . . . . 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 2861	. . . . 5 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑥 = 𝑆) → (2^nd ‘𝑦) = 𝑅)
65	56, 64	opeq12d 4813	. . . 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 2858	. . 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 220	. . . . 5 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → 𝑥:𝐶⟶(𝐴 × 𝐵))
69	68	feqmptd 6735	. . . 4 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → 𝑥 = (𝑧 ∈ 𝐶 ↦ (𝑥‘𝑧)))
70		simpr 487	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → 𝑦 = ⟨𝐷, 𝑅⟩)
71	70	fveq2d 6676	. . . . . . . . . . 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 7703	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (𝐴 ↑_m 𝐶) ∧ 𝑅 ∈ (𝐵 ↑_m 𝐶)) → (1^st ‘⟨𝐷, 𝑅⟩) = 𝐷)
75	72, 73, 74	syl2anc 586	. . . . . . . . . . 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 2858	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → (1^st ‘𝑦) = 𝐷)
77	76	fveq1d 6674	. . . . . . . . 9 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → ((1^st ‘𝑦)‘𝑧) = (𝐷‘𝑧))
78		fvex 6685	. . . . . . . . . 10 ⊢ (1^st ‘(𝑥‘𝑧)) ∈ V
79	14	fvmpt2 6781	. . . . . . . . . 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 2878	. . . . . . . 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 6676	. . . . . . . . . . 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 7704	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (𝐴 ↑_m 𝐶) ∧ 𝑅 ∈ (𝐵 ↑_m 𝐶)) → (2^nd ‘⟨𝐷, 𝑅⟩) = 𝑅)
84	72, 73, 83	syl2anc 586	. . . . . . . . . . 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 2858	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → (2^nd ‘𝑦) = 𝑅)
86	85	fveq1d 6674	. . . . . . . . 9 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → ((2^nd ‘𝑦)‘𝑧) = (𝑅‘𝑧))
87		fvex 6685	. . . . . . . . . 10 ⊢ (2^nd ‘(𝑥‘𝑧)) ∈ V
88	20	fvmpt2 6781	. . . . . . . . . 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 2878	. . . . . . . 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 4813	. . . . . . 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	ffvelrnda 6853	. . . . . . . 8 ⊢ ((((𝑥 ∈ ((𝐴 × 𝐵) ↑_m 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) ∧ 𝑧 ∈ 𝐶) → (𝑥‘𝑧) ∈ (𝐴 × 𝐵))
93		1st2nd2 7730	. . . . . . . 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 2861	. . . . . 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 5163	. . . . 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	syl5eq 2870	. . . 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 2861	. . 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 8550	1 ⊢ ((𝐴 × 𝐵) ↑_m 𝐶) ≈ ((𝐴 ↑_m 𝐶) × (𝐵 ↑_m 𝐶))

Colors of variables: wff setvar class

Syntax hints: ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3496 ⟨cop 4575 class class class wbr 5068 ↦ cmpt 5148 × cxp 5555 ⟶wf 6353 ‘cfv 6357 (class class class)co 7158 1^st c1st 7689 2^nd c2nd 7690 ↑_m cmap 8408 ≈ cen 8508

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-map 8410 df-en 8512

This theorem is referenced by: xpmapen 8687

W3C validator