Theorem xpmapenlem 6815

Description: Lemma for xpmapen 6816. (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		fnmap 6621	. . 3 ⊢ ↑𝑚 Fn (V × V)
2		xpmapen.1	. . . 4 ⊢ 𝐴 ∈ V
3		xpmapen.2	. . . 4 ⊢ 𝐵 ∈ V
4	2, 3	xpex 4719	. . 3 ⊢ (𝐴 × 𝐵) ∈ V
5		xpmapen.3	. . 3 ⊢ 𝐶 ∈ V
6		fnovex 5875	. . 3 ⊢ (( ↑𝑚 Fn (V × V) ∧ (𝐴 × 𝐵) ∈ V ∧ 𝐶 ∈ V) → ((𝐴 × 𝐵) ↑𝑚 𝐶) ∈ V)
7	1, 4, 5, 6	mp3an 1327	. 2 ⊢ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∈ V
8		fnovex 5875	. . . 4 ⊢ (( ↑𝑚 Fn (V × V) ∧ 𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴 ↑𝑚 𝐶) ∈ V)
9	1, 2, 5, 8	mp3an 1327	. . 3 ⊢ (𝐴 ↑𝑚 𝐶) ∈ V
10		fnovex 5875	. . . 4 ⊢ (( ↑𝑚 Fn (V × V) ∧ 𝐵 ∈ V ∧ 𝐶 ∈ V) → (𝐵 ↑𝑚 𝐶) ∈ V)
11	1, 3, 5, 10	mp3an 1327	. . 3 ⊢ (𝐵 ↑𝑚 𝐶) ∈ V
12	9, 11	xpex 4719	. 2 ⊢ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶)) ∈ V
13	4, 5	elmap 6643	. . . . . . 7 ⊢ (𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ↔ 𝑥:𝐶⟶(𝐴 × 𝐵))
14		ffvelrn 5618	. . . . . . 7 ⊢ ((𝑥:𝐶⟶(𝐴 × 𝐵) ∧ 𝑧 ∈ 𝐶) → (𝑥‘𝑧) ∈ (𝐴 × 𝐵))
15	13, 14	sylanb 282	. . . . . 6 ⊢ ((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑧 ∈ 𝐶) → (𝑥‘𝑧) ∈ (𝐴 × 𝐵))
16		xp1st 6133	. . . . . 6 ⊢ ((𝑥‘𝑧) ∈ (𝐴 × 𝐵) → (1st ‘(𝑥‘𝑧)) ∈ 𝐴)
17	15, 16	syl 14	. . . . 5 ⊢ ((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑧 ∈ 𝐶) → (1st ‘(𝑥‘𝑧)) ∈ 𝐴)
18		xpmapenlem.4	. . . . 5 ⊢ 𝐷 = (𝑧 ∈ 𝐶 ↦ (1st ‘(𝑥‘𝑧)))
19	17, 18	fmptd 5639	. . . 4 ⊢ (𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) → 𝐷:𝐶⟶𝐴)
20	2, 5	elmap 6643	. . . 4 ⊢ (𝐷 ∈ (𝐴 ↑𝑚 𝐶) ↔ 𝐷:𝐶⟶𝐴)
21	19, 20	sylibr 133	. . 3 ⊢ (𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) → 𝐷 ∈ (𝐴 ↑𝑚 𝐶))
22		xp2nd 6134	. . . . . 6 ⊢ ((𝑥‘𝑧) ∈ (𝐴 × 𝐵) → (2nd ‘(𝑥‘𝑧)) ∈ 𝐵)
23	15, 22	syl 14	. . . . 5 ⊢ ((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑧 ∈ 𝐶) → (2nd ‘(𝑥‘𝑧)) ∈ 𝐵)
24		xpmapenlem.5	. . . . 5 ⊢ 𝑅 = (𝑧 ∈ 𝐶 ↦ (2nd ‘(𝑥‘𝑧)))
25	23, 24	fmptd 5639	. . . 4 ⊢ (𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) → 𝑅:𝐶⟶𝐵)
26	3, 5	elmap 6643	. . . 4 ⊢ (𝑅 ∈ (𝐵 ↑𝑚 𝐶) ↔ 𝑅:𝐶⟶𝐵)
27	25, 26	sylibr 133	. . 3 ⊢ (𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) → 𝑅 ∈ (𝐵 ↑𝑚 𝐶))
28		opelxpi 4636	. . 3 ⊢ ((𝐷 ∈ (𝐴 ↑𝑚 𝐶) ∧ 𝑅 ∈ (𝐵 ↑𝑚 𝐶)) → ⟨𝐷, 𝑅⟩ ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶)))
29	21, 27, 28	syl2anc 409	. 2 ⊢ (𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) → ⟨𝐷, 𝑅⟩ ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶)))
30		xp1st 6133	. . . . . . 7 ⊢ (𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶)) → (1st ‘𝑦) ∈ (𝐴 ↑𝑚 𝐶))
31	2, 5	elmap 6643	. . . . . . 7 ⊢ ((1st ‘𝑦) ∈ (𝐴 ↑𝑚 𝐶) ↔ (1st ‘𝑦):𝐶⟶𝐴)
32	30, 31	sylib 121	. . . . . 6 ⊢ (𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶)) → (1st ‘𝑦):𝐶⟶𝐴)
33	32	ffvelrnda 5620	. . . . 5 ⊢ ((𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶)) ∧ 𝑧 ∈ 𝐶) → ((1st ‘𝑦)‘𝑧) ∈ 𝐴)
34		xp2nd 6134	. . . . . . 7 ⊢ (𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶)) → (2nd ‘𝑦) ∈ (𝐵 ↑𝑚 𝐶))
35	3, 5	elmap 6643	. . . . . . 7 ⊢ ((2nd ‘𝑦) ∈ (𝐵 ↑𝑚 𝐶) ↔ (2nd ‘𝑦):𝐶⟶𝐵)
36	34, 35	sylib 121	. . . . . 6 ⊢ (𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶)) → (2nd ‘𝑦):𝐶⟶𝐵)
37	36	ffvelrnda 5620	. . . . 5 ⊢ ((𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶)) ∧ 𝑧 ∈ 𝐶) → ((2nd ‘𝑦)‘𝑧) ∈ 𝐵)
38		opelxpi 4636	. . . . 5 ⊢ ((((1st ‘𝑦)‘𝑧) ∈ 𝐴 ∧ ((2nd ‘𝑦)‘𝑧) ∈ 𝐵) → ⟨((1st ‘𝑦)‘𝑧), ((2nd ‘𝑦)‘𝑧)⟩ ∈ (𝐴 × 𝐵))
39	33, 37, 38	syl2anc 409	. . . 4 ⊢ ((𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶)) ∧ 𝑧 ∈ 𝐶) → ⟨((1st ‘𝑦)‘𝑧), ((2nd ‘𝑦)‘𝑧)⟩ ∈ (𝐴 × 𝐵))
40		xpmapenlem.6	. . . 4 ⊢ 𝑆 = (𝑧 ∈ 𝐶 ↦ ⟨((1st ‘𝑦)‘𝑧), ((2nd ‘𝑦)‘𝑧)⟩)
41	39, 40	fmptd 5639	. . 3 ⊢ (𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶)) → 𝑆:𝐶⟶(𝐴 × 𝐵))
42	4, 5	elmap 6643	. . 3 ⊢ (𝑆 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ↔ 𝑆:𝐶⟶(𝐴 × 𝐵))
43	41, 42	sylibr 133	. 2 ⊢ (𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶)) → 𝑆 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶))
44		1st2nd2 6143	. . . . 5 ⊢ (𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶)) → 𝑦 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
45	44	ad2antlr 481	. . . 4 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑥 = 𝑆) → 𝑦 = ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩)
46	32	feqmptd 5539	. . . . . . 7 ⊢ (𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶)) → (1st ‘𝑦) = (𝑧 ∈ 𝐶 ↦ ((1st ‘𝑦)‘𝑧)))
47	46	ad2antlr 481	. . . . . 6 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑥 = 𝑆) → (1st ‘𝑦) = (𝑧 ∈ 𝐶 ↦ ((1st ‘𝑦)‘𝑧)))
48		simplr 520	. . . . . . . . . . . 12 ⊢ ((((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑥 = 𝑆) ∧ 𝑧 ∈ 𝐶) → 𝑥 = 𝑆)
49	48	fveq1d 5488	. . . . . . . . . . 11 ⊢ ((((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑥 = 𝑆) ∧ 𝑧 ∈ 𝐶) → (𝑥‘𝑧) = (𝑆‘𝑧))
50		vex 2729	. . . . . . . . . . . . . . . 16 ⊢ 𝑦 ∈ V
51		1stexg 6135	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ V → (1st ‘𝑦) ∈ V)
52	50, 51	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ (1st ‘𝑦) ∈ V
53		vex 2729	. . . . . . . . . . . . . . 15 ⊢ 𝑧 ∈ V
54	52, 53	fvex 5506	. . . . . . . . . . . . . 14 ⊢ ((1st ‘𝑦)‘𝑧) ∈ V
55		2ndexg 6136	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ V → (2nd ‘𝑦) ∈ V)
56	50, 55	ax-mp 5	. . . . . . . . . . . . . . 15 ⊢ (2nd ‘𝑦) ∈ V
57	56, 53	fvex 5506	. . . . . . . . . . . . . 14 ⊢ ((2nd ‘𝑦)‘𝑧) ∈ V
58	54, 57	opex 4207	. . . . . . . . . . . . 13 ⊢ ⟨((1st ‘𝑦)‘𝑧), ((2nd ‘𝑦)‘𝑧)⟩ ∈ V
59	40	fvmpt2 5569	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ 𝐶 ∧ ⟨((1st ‘𝑦)‘𝑧), ((2nd ‘𝑦)‘𝑧)⟩ ∈ V) → (𝑆‘𝑧) = ⟨((1st ‘𝑦)‘𝑧), ((2nd ‘𝑦)‘𝑧)⟩)
60	58, 59	mpan2 422	. . . . . . . . . . . 12 ⊢ (𝑧 ∈ 𝐶 → (𝑆‘𝑧) = ⟨((1st ‘𝑦)‘𝑧), ((2nd ‘𝑦)‘𝑧)⟩)
61	60	adantl 275	. . . . . . . . . . 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)⟩)
62	49, 61	eqtrd 2198	. . . . . . . . . 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)⟩)
63	62	fveq2d 5490	. . . . . . . . 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)⟩))
64	54, 57	op1st 6114	. . . . . . . . 9 ⊢ (1st ‘⟨((1st ‘𝑦)‘𝑧), ((2nd ‘𝑦)‘𝑧)⟩) = ((1st ‘𝑦)‘𝑧)
65	63, 64	eqtrdi 2215	. . . . . . . 8 ⊢ ((((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑥 = 𝑆) ∧ 𝑧 ∈ 𝐶) → (1st ‘(𝑥‘𝑧)) = ((1st ‘𝑦)‘𝑧))
66	65	mpteq2dva 4072	. . . . . . 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)))
67	18, 66	syl5eq 2211	. . . . . 6 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑥 = 𝑆) → 𝐷 = (𝑧 ∈ 𝐶 ↦ ((1st ‘𝑦)‘𝑧)))
68	47, 67	eqtr4d 2201	. . . . 5 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑥 = 𝑆) → (1st ‘𝑦) = 𝐷)
69	36	feqmptd 5539	. . . . . . 7 ⊢ (𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶)) → (2nd ‘𝑦) = (𝑧 ∈ 𝐶 ↦ ((2nd ‘𝑦)‘𝑧)))
70	69	ad2antlr 481	. . . . . 6 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑥 = 𝑆) → (2nd ‘𝑦) = (𝑧 ∈ 𝐶 ↦ ((2nd ‘𝑦)‘𝑧)))
71	62	fveq2d 5490	. . . . . . . . 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)⟩))
72	54, 57	op2nd 6115	. . . . . . . . 9 ⊢ (2nd ‘⟨((1st ‘𝑦)‘𝑧), ((2nd ‘𝑦)‘𝑧)⟩) = ((2nd ‘𝑦)‘𝑧)
73	71, 72	eqtrdi 2215	. . . . . . . 8 ⊢ ((((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑥 = 𝑆) ∧ 𝑧 ∈ 𝐶) → (2nd ‘(𝑥‘𝑧)) = ((2nd ‘𝑦)‘𝑧))
74	73	mpteq2dva 4072	. . . . . . 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)))
75	24, 74	syl5eq 2211	. . . . . 6 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑥 = 𝑆) → 𝑅 = (𝑧 ∈ 𝐶 ↦ ((2nd ‘𝑦)‘𝑧)))
76	70, 75	eqtr4d 2201	. . . . 5 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑥 = 𝑆) → (2nd ‘𝑦) = 𝑅)
77	68, 76	opeq12d 3766	. . . 4 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑥 = 𝑆) → ⟨(1st ‘𝑦), (2nd ‘𝑦)⟩ = ⟨𝐷, 𝑅⟩)
78	45, 77	eqtrd 2198	. . 3 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑥 = 𝑆) → 𝑦 = ⟨𝐷, 𝑅⟩)
79		simpll 519	. . . . . 6 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → 𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶))
80	79, 13	sylib 121	. . . . 5 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → 𝑥:𝐶⟶(𝐴 × 𝐵))
81	80	feqmptd 5539	. . . 4 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → 𝑥 = (𝑧 ∈ 𝐶 ↦ (𝑥‘𝑧)))
82		simpr 109	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → 𝑦 = ⟨𝐷, 𝑅⟩)
83	82	fveq2d 5490	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → (1st ‘𝑦) = (1st ‘⟨𝐷, 𝑅⟩))
84	21	ad2antrr 480	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → 𝐷 ∈ (𝐴 ↑𝑚 𝐶))
85	27	ad2antrr 480	. . . . . . . . . . . 12 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → 𝑅 ∈ (𝐵 ↑𝑚 𝐶))
86		op1stg 6118	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (𝐴 ↑𝑚 𝐶) ∧ 𝑅 ∈ (𝐵 ↑𝑚 𝐶)) → (1st ‘⟨𝐷, 𝑅⟩) = 𝐷)
87	84, 85, 86	syl2anc 409	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → (1st ‘⟨𝐷, 𝑅⟩) = 𝐷)
88	83, 87	eqtrd 2198	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → (1st ‘𝑦) = 𝐷)
89	88	fveq1d 5488	. . . . . . . . 9 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → ((1st ‘𝑦)‘𝑧) = (𝐷‘𝑧))
90		vex 2729	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
91	90, 53	fvex 5506	. . . . . . . . . . 11 ⊢ (𝑥‘𝑧) ∈ V
92		1stexg 6135	. . . . . . . . . . 11 ⊢ ((𝑥‘𝑧) ∈ V → (1st ‘(𝑥‘𝑧)) ∈ V)
93	91, 92	ax-mp 5	. . . . . . . . . 10 ⊢ (1st ‘(𝑥‘𝑧)) ∈ V
94	18	fvmpt2 5569	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝐶 ∧ (1st ‘(𝑥‘𝑧)) ∈ V) → (𝐷‘𝑧) = (1st ‘(𝑥‘𝑧)))
95	93, 94	mpan2 422	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐶 → (𝐷‘𝑧) = (1st ‘(𝑥‘𝑧)))
96	89, 95	sylan9eq 2219	. . . . . . . 8 ⊢ ((((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) ∧ 𝑧 ∈ 𝐶) → ((1st ‘𝑦)‘𝑧) = (1st ‘(𝑥‘𝑧)))
97	82	fveq2d 5490	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → (2nd ‘𝑦) = (2nd ‘⟨𝐷, 𝑅⟩))
98		op2ndg 6119	. . . . . . . . . . . 12 ⊢ ((𝐷 ∈ (𝐴 ↑𝑚 𝐶) ∧ 𝑅 ∈ (𝐵 ↑𝑚 𝐶)) → (2nd ‘⟨𝐷, 𝑅⟩) = 𝑅)
99	84, 85, 98	syl2anc 409	. . . . . . . . . . 11 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → (2nd ‘⟨𝐷, 𝑅⟩) = 𝑅)
100	97, 99	eqtrd 2198	. . . . . . . . . 10 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → (2nd ‘𝑦) = 𝑅)
101	100	fveq1d 5488	. . . . . . . . 9 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → ((2nd ‘𝑦)‘𝑧) = (𝑅‘𝑧))
102		2ndexg 6136	. . . . . . . . . . 11 ⊢ ((𝑥‘𝑧) ∈ V → (2nd ‘(𝑥‘𝑧)) ∈ V)
103	91, 102	ax-mp 5	. . . . . . . . . 10 ⊢ (2nd ‘(𝑥‘𝑧)) ∈ V
104	24	fvmpt2 5569	. . . . . . . . . 10 ⊢ ((𝑧 ∈ 𝐶 ∧ (2nd ‘(𝑥‘𝑧)) ∈ V) → (𝑅‘𝑧) = (2nd ‘(𝑥‘𝑧)))
105	103, 104	mpan2 422	. . . . . . . . 9 ⊢ (𝑧 ∈ 𝐶 → (𝑅‘𝑧) = (2nd ‘(𝑥‘𝑧)))
106	101, 105	sylan9eq 2219	. . . . . . . 8 ⊢ ((((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) ∧ 𝑧 ∈ 𝐶) → ((2nd ‘𝑦)‘𝑧) = (2nd ‘(𝑥‘𝑧)))
107	96, 106	opeq12d 3766	. . . . . . 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))⟩)
108	80	ffvelrnda 5620	. . . . . . . 8 ⊢ ((((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) ∧ 𝑧 ∈ 𝐶) → (𝑥‘𝑧) ∈ (𝐴 × 𝐵))
109		1st2nd2 6143	. . . . . . . 8 ⊢ ((𝑥‘𝑧) ∈ (𝐴 × 𝐵) → (𝑥‘𝑧) = ⟨(1st ‘(𝑥‘𝑧)), (2nd ‘(𝑥‘𝑧))⟩)
110	108, 109	syl 14	. . . . . . 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))⟩)
111	107, 110	eqtr4d 2201	. . . . . 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))
112	111	mpteq2dva 4072	. . . . 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)))
113	40, 112	syl5eq 2211	. . . 4 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → 𝑆 = (𝑧 ∈ 𝐶 ↦ (𝑥‘𝑧)))
114	81, 113	eqtr4d 2201	. . 3 ⊢ (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → 𝑥 = 𝑆)
115	78, 114	impbida 586	. 2 ⊢ ((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))) → (𝑥 = 𝑆 ↔ 𝑦 = ⟨𝐷, 𝑅⟩))
116	7, 12, 29, 43, 115	en3i 6737	1 ⊢ ((𝐴 × 𝐵) ↑𝑚 𝐶) ≈ ((𝐴 ↑𝑚 𝐶) × (𝐵 ↑𝑚 𝐶))

Syntax hints:   ∧ wa 103   = wceq 1343   ∈ wcel 2136  Vcvv 2726  ⟨cop 3579   class class class wbr 3982   ↦ cmpt 4043   × cxp 4602   Fn wfn 5183  ⟶wf 5184  ‘cfv 5188  (class class class)co 5842  1^st c1st 6106  2^nd c2nd 6107   ↑_𝑚 cmap 6614   ≈ cen 6704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-map 6616  df-en 6707

This theorem is referenced by:  xpmapen  6816