Theorem xpmapenlem 8127

Description: Lemma for xpmapen 8128. (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 ↦ (1st ‘(𝑥‘𝑧)))
xpmapenlem.5	⊢ 𝑅 = (𝑧 ∈ 𝐶 ↦ (2nd ‘(𝑥‘𝑧)))
xpmapenlem.6	⊢ 𝑆 = (𝑧 ∈ 𝐶 ↦ ⟨((1st ‘𝑦)‘𝑧), ((2nd ‘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 6678	. 2 ⊢ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∈ V
2		ovex 6678	. . 3 ⊢ (𝐴 ↑𝑚 𝐶) ∈ V
3		ovex 6678	. . 3 ⊢ (𝐵 ↑𝑚 𝐶) ∈ V
4	2, 3	xpex 6962	. 2 ⊢ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶)) ∈ V
5		xpmapen.1	. . . . . . . . 9 ⊢ 𝐴 ∈ V
6		xpmapen.2	. . . . . . . . 9 ⊢ 𝐵 ∈ V
7	5, 6	xpex 6962	. . . . . . . 8 ⊢ (𝐴 × 𝐵) ∈ V
8		xpmapen.3	. . . . . . . 8 ⊢ 𝐶 ∈ V
9	7, 8	elmap 7886	. . . . . . 7 ⊢ (𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ↔ 𝑥:𝐶⟶(𝐴 × 𝐵))
10		ffvelrn 6357	. . . . . . 7 ⊢ ((𝑥:𝐶⟶(𝐴 × 𝐵) ∧ 𝑧 ∈ 𝐶) → (𝑥‘𝑧) ∈ (𝐴 × 𝐵))
11	9, 10	sylanb 489	. . . . . 6 ⊢ ((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑧 ∈ 𝐶) → (𝑥‘𝑧) ∈ (𝐴 × 𝐵))
12		xp1st 7198	. . . . . 6 ⊢ ((𝑥‘𝑧) ∈ (𝐴 × 𝐵) → (1st ‘(𝑥‘𝑧)) ∈ 𝐴)
13	11, 12	syl 17	. . . . 5 ⊢ ((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑧 ∈ 𝐶) → (1st ‘(𝑥‘𝑧)) ∈ 𝐴)
14		xpmapenlem.4	. . . . 5 ⊢ 𝐷 = (𝑧 ∈ 𝐶 ↦ (1st ‘(𝑥‘𝑧)))
15	13, 14	fmptd 6385	. . . 4 ⊢ (𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) → 𝐷:𝐶⟶𝐴)
16	5, 8	elmap 7886	. . . 4 ⊢ (𝐷 ∈ (𝐴 ↑𝑚 𝐶) ↔ 𝐷:𝐶⟶𝐴)
17	15, 16	sylibr 224	. . 3 ⊢ (𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) → 𝐷 ∈ (𝐴 ↑𝑚 𝐶))
18		xp2nd 7199	. . . . . 6 ⊢ ((𝑥‘𝑧) ∈ (𝐴 × 𝐵) → (2nd ‘(𝑥‘𝑧)) ∈ 𝐵)
19	11, 18	syl 17	. . . . 5 ⊢ ((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑧 ∈ 𝐶) → (2nd ‘(𝑥‘𝑧)) ∈ 𝐵)
20		xpmapenlem.5	. . . . 5 ⊢ 𝑅 = (𝑧 ∈ 𝐶 ↦ (2nd ‘(𝑥‘𝑧)))
21	19, 20	fmptd 6385	. . . 4 ⊢ (𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) → 𝑅:𝐶⟶𝐵)
22	6, 8	elmap 7886	. . . 4 ⊢ (𝑅 ∈ (𝐵 ↑𝑚 𝐶) ↔ 𝑅:𝐶⟶𝐵)
23	21, 22	sylibr 224	. . 3 ⊢ (𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) → 𝑅 ∈ (𝐵 ↑𝑚 𝐶))
24		opelxpi 5148	. . 3 ⊢ ((𝐷 ∈ (𝐴 ↑𝑚 𝐶) ∧ 𝑅 ∈ (𝐵 ↑𝑚 𝐶)) → ⟨𝐷, 𝑅⟩ ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶)))
25	17, 23, 24	syl2anc 693	. 2 ⊢ (𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) → ⟨𝐷, 𝑅⟩ ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶)))
26		xp1st 7198	. . . . . . 7 ⊢ (𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶)) → (1st ‘𝑦) ∈ (𝐴 ↑𝑚 𝐶))
27	5, 8	elmap 7886	. . . . . . 7 ⊢ ((1st ‘𝑦) ∈ (𝐴 ↑𝑚 𝐶) ↔ (1st ‘𝑦):𝐶⟶𝐴)
28	26, 27	sylib 208	. . . . . 6 ⊢ (𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶)) → (1st ‘𝑦):𝐶⟶𝐴)
29	28	ffvelrnda 6359	. . . . 5 ⊢ ((𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶)) ∧ 𝑧 ∈ 𝐶) → ((1st ‘𝑦)‘𝑧) ∈ 𝐴)
30		xp2nd 7199	. . . . . . 7 ⊢ (𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶)) → (2nd ‘𝑦) ∈ (𝐵 ↑𝑚 𝐶))
31	6, 8	elmap 7886	. . . . . . 7 ⊢ ((2nd ‘𝑦) ∈ (𝐵 ↑𝑚 𝐶) ↔ (2nd ‘𝑦):𝐶⟶𝐵)
32	30, 31	sylib 208	. . . . . 6 ⊢ (𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶)) → (2nd ‘𝑦):𝐶⟶𝐵)
33	32	ffvelrnda 6359	. . . . 5 ⊢ ((𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶)) ∧ 𝑧 ∈ 𝐶) → ((2nd ‘𝑦)‘𝑧) ∈ 𝐵)
34		opelxpi 5148	. . . . 5 ⊢ ((((1st ‘𝑦)‘𝑧) ∈ 𝐴 ∧ ((2nd ‘𝑦)‘𝑧) ∈ 𝐵) → ⟨((1st ‘𝑦)‘𝑧), ((2nd ‘𝑦)‘𝑧)⟩ ∈ (𝐴 × 𝐵))
35	29, 33, 34	syl2anc 693	. . . 4 ⊢ ((𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶)) ∧ 𝑧 ∈ 𝐶) → ⟨((1st ‘𝑦)‘𝑧), ((2nd ‘𝑦)‘𝑧)⟩ ∈ (𝐴 × 𝐵))
36		xpmapenlem.6	. . . 4 ⊢ 𝑆 = (𝑧 ∈ 𝐶 ↦ ⟨((1st ‘𝑦)‘𝑧), ((2nd ‘𝑦)‘𝑧)⟩)
37	35, 36	fmptd 6385	. . 3 ⊢ (𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶)) → 𝑆:𝐶⟶(𝐴 × 𝐵))
38	7, 8	elmap 7886	. . 3 ⊢ (𝑆 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ↔ 𝑆:𝐶⟶(𝐴 × 𝐵))
39	37, 38	sylibr 224	. 2 ⊢ (𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶)) → 𝑆 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶))
40		1st2nd2 7205	. . . . 5 ⊢ (𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶)) → 𝑦 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
41	40	ad2antlr 763	. . . 4 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑥 = 𝑆) → 𝑦 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
42	28	feqmptd 6249	. . . . . . 7 ⊢ (𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶)) → (1st ‘𝑦) = (𝑧 ∈ 𝐶 ↦ ((1st ‘𝑦)‘𝑧)))
43	42	ad2antlr 763	. . . . . 6 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑥 = 𝑆) → (1st ‘𝑦) = (𝑧 ∈ 𝐶 ↦ ((1st ‘𝑦)‘𝑧)))
44		simplr 792	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑥 = 𝑆) ∧ 𝑧 ∈ 𝐶) → 𝑥 = 𝑆)
45	44	fveq1d 6193	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑥 = 𝑆) ∧ 𝑧 ∈ 𝐶) → (𝑥‘𝑧) = (𝑆‘𝑧))
46		opex 4932	. . . . . . . . . . . . 13 ⊢ ⟨((1st ‘𝑦)‘𝑧), ((2nd ‘𝑦)‘𝑧)⟩ ∈ V
47	36	fvmpt2 6291	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ 𝐶 ∧ ⟨((1st ‘𝑦)‘𝑧), ((2nd ‘𝑦)‘𝑧)⟩ ∈ V) → (𝑆‘𝑧) = ⟨((1st ‘𝑦)‘𝑧), ((2nd ‘𝑦)‘𝑧)⟩)
48	46, 47	mpan2 707	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝐶 → (𝑆‘𝑧) = ⟨((1st ‘𝑦)‘𝑧), ((2nd ‘𝑦)‘𝑧)⟩)
49	48	adantl 482	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑥 = 𝑆) ∧ 𝑧 ∈ 𝐶) → (𝑆‘𝑧) = ⟨((1st ‘𝑦)‘𝑧), ((2nd ‘𝑦)‘𝑧)⟩)
50	45, 49	eqtrd 2656	. . . . . . . . . 10 ⊢ ((((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑥 = 𝑆) ∧ 𝑧 ∈ 𝐶) → (𝑥‘𝑧) = ⟨((1st ‘𝑦)‘𝑧), ((2nd ‘𝑦)‘𝑧)⟩)
51	50	fveq2d 6195	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑥 = 𝑆) ∧ 𝑧 ∈ 𝐶) → (1st ‘(𝑥‘𝑧)) = (1st ‘⟨((1st ‘𝑦)‘𝑧), ((2nd ‘𝑦)‘𝑧)⟩))
52		fvex 6201	. . . . . . . . . 10 ⊢ ((1st ‘𝑦)‘𝑧) ∈ V
53		fvex 6201	. . . . . . . . . 10 ⊢ ((2nd ‘𝑦)‘𝑧) ∈ V
54	52, 53	op1st 7176	. . . . . . . . 9 ⊢ (1st ‘⟨((1st ‘𝑦)‘𝑧), ((2nd ‘𝑦)‘𝑧)⟩) = ((1st ‘𝑦)‘𝑧)
55	51, 54	syl6eq 2672	. . . . . . . 8 ⊢ ((((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑥 = 𝑆) ∧ 𝑧 ∈ 𝐶) → (1st ‘(𝑥‘𝑧)) = ((1st ‘𝑦)‘𝑧))
56	55	mpteq2dva 4744	. . . . . . 7 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑥 = 𝑆) → (𝑧 ∈ 𝐶 ↦ (1st ‘(𝑥‘𝑧))) = (𝑧 ∈ 𝐶 ↦ ((1st ‘𝑦)‘𝑧)))
57	14, 56	syl5eq 2668	. . . . . 6 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑥 = 𝑆) → 𝐷 = (𝑧 ∈ 𝐶 ↦ ((1st ‘𝑦)‘𝑧)))
58	43, 57	eqtr4d 2659	. . . . 5 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑥 = 𝑆) → (1st ‘𝑦) = 𝐷)
59	32	feqmptd 6249	. . . . . . 7 ⊢ (𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶)) → (2nd ‘𝑦) = (𝑧 ∈ 𝐶 ↦ ((2nd ‘𝑦)‘𝑧)))
60	59	ad2antlr 763	. . . . . 6 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑥 = 𝑆) → (2nd ‘𝑦) = (𝑧 ∈ 𝐶 ↦ ((2nd ‘𝑦)‘𝑧)))
61	50	fveq2d 6195	. . . . . . . . 9 ⊢ ((((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑥 = 𝑆) ∧ 𝑧 ∈ 𝐶) → (2nd ‘(𝑥‘𝑧)) = (2nd ‘⟨((1st ‘𝑦)‘𝑧), ((2nd ‘𝑦)‘𝑧)⟩))
62	52, 53	op2nd 7177	. . . . . . . . 9 ⊢ (2nd ‘⟨((1st ‘𝑦)‘𝑧), ((2nd ‘𝑦)‘𝑧)⟩) = ((2nd ‘𝑦)‘𝑧)
63	61, 62	syl6eq 2672	. . . . . . . 8 ⊢ ((((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑥 = 𝑆) ∧ 𝑧 ∈ 𝐶) → (2nd ‘(𝑥‘𝑧)) = ((2nd ‘𝑦)‘𝑧))
64	63	mpteq2dva 4744	. . . . . . 7 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑥 = 𝑆) → (𝑧 ∈ 𝐶 ↦ (2nd ‘(𝑥‘𝑧))) = (𝑧 ∈ 𝐶 ↦ ((2nd ‘𝑦)‘𝑧)))
65	20, 64	syl5eq 2668	. . . . . 6 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑥 = 𝑆) → 𝑅 = (𝑧 ∈ 𝐶 ↦ ((2nd ‘𝑦)‘𝑧)))
66	60, 65	eqtr4d 2659	. . . . 5 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑥 = 𝑆) → (2nd ‘𝑦) = 𝑅)
67	58, 66	opeq12d 4410	. . . 4 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑥 = 𝑆) → ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ = ⟨𝐷, 𝑅⟩)
68	41, 67	eqtrd 2656	. . 3 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑥 = 𝑆) → 𝑦 = ⟨𝐷, 𝑅⟩)
69		simpll 790	. . . . . 6 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → 𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶))
70	69, 9	sylib 208	. . . . 5 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → 𝑥:𝐶⟶(𝐴 × 𝐵))
71	70	feqmptd 6249	. . . 4 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → 𝑥 = (𝑧 ∈ 𝐶 ↦ (𝑥‘𝑧)))
72		simpr 477	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → 𝑦 = ⟨𝐷, 𝑅⟩)
73	72	fveq2d 6195	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → (1st ‘𝑦) = (1st ‘⟨𝐷, 𝑅⟩))
74	17	ad2antrr 762	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → 𝐷 ∈ (𝐴 ↑𝑚 𝐶))
75	23	ad2antrr 762	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → 𝑅 ∈ (𝐵 ↑𝑚 𝐶))
76		op1stg 7180	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (𝐴 ↑𝑚 𝐶) ∧ 𝑅 ∈ (𝐵 ↑𝑚 𝐶)) → (1st ‘⟨𝐷, 𝑅⟩) = 𝐷)
77	74, 75, 76	syl2anc 693	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → (1st ‘⟨𝐷, 𝑅⟩) = 𝐷)
78	73, 77	eqtrd 2656	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → (1st ‘𝑦) = 𝐷)
79	78	fveq1d 6193	. . . . . . . . 9 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → ((1st ‘𝑦)‘𝑧) = (𝐷‘𝑧))
80		fvex 6201	. . . . . . . . . 10 ⊢ (1st ‘(𝑥‘𝑧)) ∈ V
81	14	fvmpt2 6291	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝐶 ∧ (1st ‘(𝑥‘𝑧)) ∈ V) → (𝐷‘𝑧) = (1st ‘(𝑥‘𝑧)))
82	80, 81	mpan2 707	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐶 → (𝐷‘𝑧) = (1st ‘(𝑥‘𝑧)))
83	79, 82	sylan9eq 2676	. . . . . . . 8 ⊢ ((((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) ∧ 𝑧 ∈ 𝐶) → ((1st ‘𝑦)‘𝑧) = (1st ‘(𝑥‘𝑧)))
84	72	fveq2d 6195	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → (2nd ‘𝑦) = (2nd ‘⟨𝐷, 𝑅⟩))
85		op2ndg 7181	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (𝐴 ↑𝑚 𝐶) ∧ 𝑅 ∈ (𝐵 ↑𝑚 𝐶)) → (2nd ‘⟨𝐷, 𝑅⟩) = 𝑅)
86	74, 75, 85	syl2anc 693	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → (2nd ‘⟨𝐷, 𝑅⟩) = 𝑅)
87	84, 86	eqtrd 2656	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → (2nd ‘𝑦) = 𝑅)
88	87	fveq1d 6193	. . . . . . . . 9 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → ((2nd ‘𝑦)‘𝑧) = (𝑅‘𝑧))
89		fvex 6201	. . . . . . . . . 10 ⊢ (2nd ‘(𝑥‘𝑧)) ∈ V
90	20	fvmpt2 6291	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝐶 ∧ (2nd ‘(𝑥‘𝑧)) ∈ V) → (𝑅‘𝑧) = (2nd ‘(𝑥‘𝑧)))
91	89, 90	mpan2 707	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐶 → (𝑅‘𝑧) = (2nd ‘(𝑥‘𝑧)))
92	88, 91	sylan9eq 2676	. . . . . . . 8 ⊢ ((((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) ∧ 𝑧 ∈ 𝐶) → ((2nd ‘𝑦)‘𝑧) = (2nd ‘(𝑥‘𝑧)))
93	83, 92	opeq12d 4410	. . . . . . 7 ⊢ ((((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) ∧ 𝑧 ∈ 𝐶) → ⟨((1st ‘𝑦)‘𝑧), ((2nd ‘𝑦)‘𝑧)⟩ = ⟨(1st ‘(𝑥‘𝑧)), (2nd ‘(𝑥‘𝑧))⟩)
94	70	ffvelrnda 6359	. . . . . . . 8 ⊢ ((((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) ∧ 𝑧 ∈ 𝐶) → (𝑥‘𝑧) ∈ (𝐴 × 𝐵))
95		1st2nd2 7205	. . . . . . . 8 ⊢ ((𝑥‘𝑧) ∈ (𝐴 × 𝐵) → (𝑥‘𝑧) = ⟨(1st ‘(𝑥‘𝑧)), (2nd ‘(𝑥‘𝑧))⟩)
96	94, 95	syl 17	. . . . . . 7 ⊢ ((((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) ∧ 𝑧 ∈ 𝐶) → (𝑥‘𝑧) = ⟨(1st ‘(𝑥‘𝑧)), (2nd ‘(𝑥‘𝑧))⟩)
97	93, 96	eqtr4d 2659	. . . . . 6 ⊢ ((((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) ∧ 𝑧 ∈ 𝐶) → ⟨((1st ‘𝑦)‘𝑧), ((2nd ‘𝑦)‘𝑧)⟩ = (𝑥‘𝑧))
98	97	mpteq2dva 4744	. . . . 5 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → (𝑧 ∈ 𝐶 ↦ ⟨((1st ‘𝑦)‘𝑧), ((2nd ‘𝑦)‘𝑧)⟩) = (𝑧 ∈ 𝐶 ↦ (𝑥‘𝑧)))
99	36, 98	syl5eq 2668	. . . 4 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → 𝑆 = (𝑧 ∈ 𝐶 ↦ (𝑥‘𝑧)))
100	71, 99	eqtr4d 2659	. . 3 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → 𝑥 = 𝑆)
101	68, 100	impbida 877	. 2 ⊢ ((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) → (𝑥 = 𝑆 ↔ 𝑦 = ⟨𝐷, 𝑅⟩))
102	1, 4, 25, 39, 101	en3i 7994	1 ⊢ ((𝐴 × 𝐵) ↑𝑚 𝐶) ≈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))

Syntax hints:   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200  ⟨cop 4183   class class class wbr 4653   ↦ cmpt 4729   × cxp 5112  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  1^st c1st 7166  2^nd c2nd 7167   ↑_𝑚 cmap 7857   ≈ cen 7952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859  df-en 7956

This theorem is referenced by:  xpmapen  8128