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Theorem xpmapenlem 8660
Description: Lemma for xpmapen 8661. (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 = (𝑧𝐶 ↦ (1st ‘(𝑥𝑧)))
xpmapenlem.5 𝑅 = (𝑧𝐶 ↦ (2nd ‘(𝑥𝑧)))
xpmapenlem.6 𝑆 = (𝑧𝐶 ↦ ⟨((1st𝑦)‘𝑧), ((2nd𝑦)‘𝑧)⟩)
Assertion
Ref Expression
xpmapenlem ((𝐴 × 𝐵) ↑m 𝐶) ≈ ((𝐴m 𝐶) × (𝐵m 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
StepHypRef Expression
1 ovex 7163 . 2 ((𝐴 × 𝐵) ↑m 𝐶) ∈ V
2 ovex 7163 . . 3 (𝐴m 𝐶) ∈ V
3 ovex 7163 . . 3 (𝐵m 𝐶) ∈ V
42, 3xpex 7451 . 2 ((𝐴m 𝐶) × (𝐵m 𝐶)) ∈ V
5 xpmapen.1 . . . . . . . . 9 𝐴 ∈ V
6 xpmapen.2 . . . . . . . . 9 𝐵 ∈ V
75, 6xpex 7451 . . . . . . . 8 (𝐴 × 𝐵) ∈ V
8 xpmapen.3 . . . . . . . 8 𝐶 ∈ V
97, 8elmap 8410 . . . . . . 7 (𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ↔ 𝑥:𝐶⟶(𝐴 × 𝐵))
10 ffvelrn 6822 . . . . . . 7 ((𝑥:𝐶⟶(𝐴 × 𝐵) ∧ 𝑧𝐶) → (𝑥𝑧) ∈ (𝐴 × 𝐵))
119, 10sylanb 584 . . . . . 6 ((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑧𝐶) → (𝑥𝑧) ∈ (𝐴 × 𝐵))
12 xp1st 7696 . . . . . 6 ((𝑥𝑧) ∈ (𝐴 × 𝐵) → (1st ‘(𝑥𝑧)) ∈ 𝐴)
1311, 12syl 17 . . . . 5 ((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑧𝐶) → (1st ‘(𝑥𝑧)) ∈ 𝐴)
14 xpmapenlem.4 . . . . 5 𝐷 = (𝑧𝐶 ↦ (1st ‘(𝑥𝑧)))
1513, 14fmptd 6851 . . . 4 (𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) → 𝐷:𝐶𝐴)
165, 8elmap 8410 . . . 4 (𝐷 ∈ (𝐴m 𝐶) ↔ 𝐷:𝐶𝐴)
1715, 16sylibr 237 . . 3 (𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) → 𝐷 ∈ (𝐴m 𝐶))
18 xp2nd 7697 . . . . . 6 ((𝑥𝑧) ∈ (𝐴 × 𝐵) → (2nd ‘(𝑥𝑧)) ∈ 𝐵)
1911, 18syl 17 . . . . 5 ((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑧𝐶) → (2nd ‘(𝑥𝑧)) ∈ 𝐵)
20 xpmapenlem.5 . . . . 5 𝑅 = (𝑧𝐶 ↦ (2nd ‘(𝑥𝑧)))
2119, 20fmptd 6851 . . . 4 (𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) → 𝑅:𝐶𝐵)
226, 8elmap 8410 . . . 4 (𝑅 ∈ (𝐵m 𝐶) ↔ 𝑅:𝐶𝐵)
2321, 22sylibr 237 . . 3 (𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) → 𝑅 ∈ (𝐵m 𝐶))
2417, 23opelxpd 5566 . 2 (𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) → ⟨𝐷, 𝑅⟩ ∈ ((𝐴m 𝐶) × (𝐵m 𝐶)))
25 xp1st 7696 . . . . . . 7 (𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶)) → (1st𝑦) ∈ (𝐴m 𝐶))
265, 8elmap 8410 . . . . . . 7 ((1st𝑦) ∈ (𝐴m 𝐶) ↔ (1st𝑦):𝐶𝐴)
2725, 26sylib 221 . . . . . 6 (𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶)) → (1st𝑦):𝐶𝐴)
2827ffvelrnda 6824 . . . . 5 ((𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶)) ∧ 𝑧𝐶) → ((1st𝑦)‘𝑧) ∈ 𝐴)
29 xp2nd 7697 . . . . . . 7 (𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶)) → (2nd𝑦) ∈ (𝐵m 𝐶))
306, 8elmap 8410 . . . . . . 7 ((2nd𝑦) ∈ (𝐵m 𝐶) ↔ (2nd𝑦):𝐶𝐵)
3129, 30sylib 221 . . . . . 6 (𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶)) → (2nd𝑦):𝐶𝐵)
3231ffvelrnda 6824 . . . . 5 ((𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶)) ∧ 𝑧𝐶) → ((2nd𝑦)‘𝑧) ∈ 𝐵)
3328, 32opelxpd 5566 . . . 4 ((𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶)) ∧ 𝑧𝐶) → ⟨((1st𝑦)‘𝑧), ((2nd𝑦)‘𝑧)⟩ ∈ (𝐴 × 𝐵))
34 xpmapenlem.6 . . . 4 𝑆 = (𝑧𝐶 ↦ ⟨((1st𝑦)‘𝑧), ((2nd𝑦)‘𝑧)⟩)
3533, 34fmptd 6851 . . 3 (𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶)) → 𝑆:𝐶⟶(𝐴 × 𝐵))
367, 8elmap 8410 . . 3 (𝑆 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ↔ 𝑆:𝐶⟶(𝐴 × 𝐵))
3735, 36sylibr 237 . 2 (𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶)) → 𝑆 ∈ ((𝐴 × 𝐵) ↑m 𝐶))
38 1st2nd2 7703 . . . . 5 (𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶)) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
3938ad2antlr 726 . . . 4 (((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑥 = 𝑆) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
4027feqmptd 6706 . . . . . . 7 (𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶)) → (1st𝑦) = (𝑧𝐶 ↦ ((1st𝑦)‘𝑧)))
4140ad2antlr 726 . . . . . 6 (((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑥 = 𝑆) → (1st𝑦) = (𝑧𝐶 ↦ ((1st𝑦)‘𝑧)))
42 simplr 768 . . . . . . . . . . . 12 ((((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑥 = 𝑆) ∧ 𝑧𝐶) → 𝑥 = 𝑆)
4342fveq1d 6645 . . . . . . . . . . 11 ((((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑥 = 𝑆) ∧ 𝑧𝐶) → (𝑥𝑧) = (𝑆𝑧))
44 opex 5329 . . . . . . . . . . . . 13 ⟨((1st𝑦)‘𝑧), ((2nd𝑦)‘𝑧)⟩ ∈ V
4534fvmpt2 6752 . . . . . . . . . . . . 13 ((𝑧𝐶 ∧ ⟨((1st𝑦)‘𝑧), ((2nd𝑦)‘𝑧)⟩ ∈ V) → (𝑆𝑧) = ⟨((1st𝑦)‘𝑧), ((2nd𝑦)‘𝑧)⟩)
4644, 45mpan2 690 . . . . . . . . . . . 12 (𝑧𝐶 → (𝑆𝑧) = ⟨((1st𝑦)‘𝑧), ((2nd𝑦)‘𝑧)⟩)
4746adantl 485 . . . . . . . . . . 11 ((((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑥 = 𝑆) ∧ 𝑧𝐶) → (𝑆𝑧) = ⟨((1st𝑦)‘𝑧), ((2nd𝑦)‘𝑧)⟩)
4843, 47eqtrd 2856 . . . . . . . . . 10 ((((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑥 = 𝑆) ∧ 𝑧𝐶) → (𝑥𝑧) = ⟨((1st𝑦)‘𝑧), ((2nd𝑦)‘𝑧)⟩)
4948fveq2d 6647 . . . . . . . . 9 ((((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑥 = 𝑆) ∧ 𝑧𝐶) → (1st ‘(𝑥𝑧)) = (1st ‘⟨((1st𝑦)‘𝑧), ((2nd𝑦)‘𝑧)⟩))
50 fvex 6656 . . . . . . . . . 10 ((1st𝑦)‘𝑧) ∈ V
51 fvex 6656 . . . . . . . . . 10 ((2nd𝑦)‘𝑧) ∈ V
5250, 51op1st 7672 . . . . . . . . 9 (1st ‘⟨((1st𝑦)‘𝑧), ((2nd𝑦)‘𝑧)⟩) = ((1st𝑦)‘𝑧)
5349, 52syl6eq 2872 . . . . . . . 8 ((((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑥 = 𝑆) ∧ 𝑧𝐶) → (1st ‘(𝑥𝑧)) = ((1st𝑦)‘𝑧))
5453mpteq2dva 5134 . . . . . . 7 (((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑥 = 𝑆) → (𝑧𝐶 ↦ (1st ‘(𝑥𝑧))) = (𝑧𝐶 ↦ ((1st𝑦)‘𝑧)))
5514, 54syl5eq 2868 . . . . . 6 (((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑥 = 𝑆) → 𝐷 = (𝑧𝐶 ↦ ((1st𝑦)‘𝑧)))
5641, 55eqtr4d 2859 . . . . 5 (((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑥 = 𝑆) → (1st𝑦) = 𝐷)
5731feqmptd 6706 . . . . . . 7 (𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶)) → (2nd𝑦) = (𝑧𝐶 ↦ ((2nd𝑦)‘𝑧)))
5857ad2antlr 726 . . . . . 6 (((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑥 = 𝑆) → (2nd𝑦) = (𝑧𝐶 ↦ ((2nd𝑦)‘𝑧)))
5948fveq2d 6647 . . . . . . . . 9 ((((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑥 = 𝑆) ∧ 𝑧𝐶) → (2nd ‘(𝑥𝑧)) = (2nd ‘⟨((1st𝑦)‘𝑧), ((2nd𝑦)‘𝑧)⟩))
6050, 51op2nd 7673 . . . . . . . . 9 (2nd ‘⟨((1st𝑦)‘𝑧), ((2nd𝑦)‘𝑧)⟩) = ((2nd𝑦)‘𝑧)
6159, 60syl6eq 2872 . . . . . . . 8 ((((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑥 = 𝑆) ∧ 𝑧𝐶) → (2nd ‘(𝑥𝑧)) = ((2nd𝑦)‘𝑧))
6261mpteq2dva 5134 . . . . . . 7 (((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑥 = 𝑆) → (𝑧𝐶 ↦ (2nd ‘(𝑥𝑧))) = (𝑧𝐶 ↦ ((2nd𝑦)‘𝑧)))
6320, 62syl5eq 2868 . . . . . 6 (((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑥 = 𝑆) → 𝑅 = (𝑧𝐶 ↦ ((2nd𝑦)‘𝑧)))
6458, 63eqtr4d 2859 . . . . 5 (((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑥 = 𝑆) → (2nd𝑦) = 𝑅)
6556, 64opeq12d 4784 . . . 4 (((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑥 = 𝑆) → ⟨(1st𝑦), (2nd𝑦)⟩ = ⟨𝐷, 𝑅⟩)
6639, 65eqtrd 2856 . . 3 (((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑥 = 𝑆) → 𝑦 = ⟨𝐷, 𝑅⟩)
67 simpll 766 . . . . . 6 (((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → 𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶))
6867, 9sylib 221 . . . . 5 (((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → 𝑥:𝐶⟶(𝐴 × 𝐵))
6968feqmptd 6706 . . . 4 (((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → 𝑥 = (𝑧𝐶 ↦ (𝑥𝑧)))
70 simpr 488 . . . . . . . . . . . 12 (((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → 𝑦 = ⟨𝐷, 𝑅⟩)
7170fveq2d 6647 . . . . . . . . . . 11 (((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → (1st𝑦) = (1st ‘⟨𝐷, 𝑅⟩))
7217ad2antrr 725 . . . . . . . . . . . 12 (((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → 𝐷 ∈ (𝐴m 𝐶))
7323ad2antrr 725 . . . . . . . . . . . 12 (((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → 𝑅 ∈ (𝐵m 𝐶))
74 op1stg 7676 . . . . . . . . . . . 12 ((𝐷 ∈ (𝐴m 𝐶) ∧ 𝑅 ∈ (𝐵m 𝐶)) → (1st ‘⟨𝐷, 𝑅⟩) = 𝐷)
7572, 73, 74syl2anc 587 . . . . . . . . . . 11 (((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → (1st ‘⟨𝐷, 𝑅⟩) = 𝐷)
7671, 75eqtrd 2856 . . . . . . . . . 10 (((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → (1st𝑦) = 𝐷)
7776fveq1d 6645 . . . . . . . . 9 (((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → ((1st𝑦)‘𝑧) = (𝐷𝑧))
78 fvex 6656 . . . . . . . . . 10 (1st ‘(𝑥𝑧)) ∈ V
7914fvmpt2 6752 . . . . . . . . . 10 ((𝑧𝐶 ∧ (1st ‘(𝑥𝑧)) ∈ V) → (𝐷𝑧) = (1st ‘(𝑥𝑧)))
8078, 79mpan2 690 . . . . . . . . 9 (𝑧𝐶 → (𝐷𝑧) = (1st ‘(𝑥𝑧)))
8177, 80sylan9eq 2876 . . . . . . . 8 ((((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) ∧ 𝑧𝐶) → ((1st𝑦)‘𝑧) = (1st ‘(𝑥𝑧)))
8270fveq2d 6647 . . . . . . . . . . 11 (((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → (2nd𝑦) = (2nd ‘⟨𝐷, 𝑅⟩))
83 op2ndg 7677 . . . . . . . . . . . 12 ((𝐷 ∈ (𝐴m 𝐶) ∧ 𝑅 ∈ (𝐵m 𝐶)) → (2nd ‘⟨𝐷, 𝑅⟩) = 𝑅)
8472, 73, 83syl2anc 587 . . . . . . . . . . 11 (((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → (2nd ‘⟨𝐷, 𝑅⟩) = 𝑅)
8582, 84eqtrd 2856 . . . . . . . . . 10 (((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → (2nd𝑦) = 𝑅)
8685fveq1d 6645 . . . . . . . . 9 (((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → ((2nd𝑦)‘𝑧) = (𝑅𝑧))
87 fvex 6656 . . . . . . . . . 10 (2nd ‘(𝑥𝑧)) ∈ V
8820fvmpt2 6752 . . . . . . . . . 10 ((𝑧𝐶 ∧ (2nd ‘(𝑥𝑧)) ∈ V) → (𝑅𝑧) = (2nd ‘(𝑥𝑧)))
8987, 88mpan2 690 . . . . . . . . 9 (𝑧𝐶 → (𝑅𝑧) = (2nd ‘(𝑥𝑧)))
9086, 89sylan9eq 2876 . . . . . . . 8 ((((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) ∧ 𝑧𝐶) → ((2nd𝑦)‘𝑧) = (2nd ‘(𝑥𝑧)))
9181, 90opeq12d 4784 . . . . . . 7 ((((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) ∧ 𝑧𝐶) → ⟨((1st𝑦)‘𝑧), ((2nd𝑦)‘𝑧)⟩ = ⟨(1st ‘(𝑥𝑧)), (2nd ‘(𝑥𝑧))⟩)
9268ffvelrnda 6824 . . . . . . . 8 ((((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) ∧ 𝑧𝐶) → (𝑥𝑧) ∈ (𝐴 × 𝐵))
93 1st2nd2 7703 . . . . . . . 8 ((𝑥𝑧) ∈ (𝐴 × 𝐵) → (𝑥𝑧) = ⟨(1st ‘(𝑥𝑧)), (2nd ‘(𝑥𝑧))⟩)
9492, 93syl 17 . . . . . . 7 ((((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) ∧ 𝑧𝐶) → (𝑥𝑧) = ⟨(1st ‘(𝑥𝑧)), (2nd ‘(𝑥𝑧))⟩)
9591, 94eqtr4d 2859 . . . . . 6 ((((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) ∧ 𝑧𝐶) → ⟨((1st𝑦)‘𝑧), ((2nd𝑦)‘𝑧)⟩ = (𝑥𝑧))
9695mpteq2dva 5134 . . . . 5 (((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → (𝑧𝐶 ↦ ⟨((1st𝑦)‘𝑧), ((2nd𝑦)‘𝑧)⟩) = (𝑧𝐶 ↦ (𝑥𝑧)))
9734, 96syl5eq 2868 . . . 4 (((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → 𝑆 = (𝑧𝐶 ↦ (𝑥𝑧)))
9869, 97eqtr4d 2859 . . 3 (((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → 𝑥 = 𝑆)
9966, 98impbida 800 . 2 ((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) → (𝑥 = 𝑆𝑦 = ⟨𝐷, 𝑅⟩))
1001, 4, 24, 37, 99en3i 8523 1 ((𝐴 × 𝐵) ↑m 𝐶) ≈ ((𝐴m 𝐶) × (𝐵m 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wa 399   = wceq 1538  wcel 2115  Vcvv 3471  cop 4546   class class class wbr 5039  cmpt 5119   × cxp 5526  wf 6324  cfv 6328  (class class class)co 7130  1st c1st 7662  2nd c2nd 7663  m cmap 8381  cen 8481
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-uni 4812  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135  df-1st 7664  df-2nd 7665  df-map 8383  df-en 8485
This theorem is referenced by:  xpmapen  8661
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