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Theorem xpmapenlem 8931
Description: Lemma for xpmapen 8932. (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 7308 . 2 ((𝐴 × 𝐵) ↑m 𝐶) ∈ V
2 ovex 7308 . . 3 (𝐴m 𝐶) ∈ V
3 ovex 7308 . . 3 (𝐵m 𝐶) ∈ V
42, 3xpex 7603 . 2 ((𝐴m 𝐶) × (𝐵m 𝐶)) ∈ V
5 xpmapen.1 . . . . . . . . 9 𝐴 ∈ V
6 xpmapen.2 . . . . . . . . 9 𝐵 ∈ V
75, 6xpex 7603 . . . . . . . 8 (𝐴 × 𝐵) ∈ V
8 xpmapen.3 . . . . . . . 8 𝐶 ∈ V
97, 8elmap 8659 . . . . . . 7 (𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ↔ 𝑥:𝐶⟶(𝐴 × 𝐵))
10 ffvelrn 6959 . . . . . . 7 ((𝑥:𝐶⟶(𝐴 × 𝐵) ∧ 𝑧𝐶) → (𝑥𝑧) ∈ (𝐴 × 𝐵))
119, 10sylanb 581 . . . . . 6 ((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑧𝐶) → (𝑥𝑧) ∈ (𝐴 × 𝐵))
12 xp1st 7863 . . . . . 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 6988 . . . 4 (𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) → 𝐷:𝐶𝐴)
165, 8elmap 8659 . . . 4 (𝐷 ∈ (𝐴m 𝐶) ↔ 𝐷:𝐶𝐴)
1715, 16sylibr 233 . . 3 (𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) → 𝐷 ∈ (𝐴m 𝐶))
18 xp2nd 7864 . . . . . 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 6988 . . . 4 (𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) → 𝑅:𝐶𝐵)
226, 8elmap 8659 . . . 4 (𝑅 ∈ (𝐵m 𝐶) ↔ 𝑅:𝐶𝐵)
2321, 22sylibr 233 . . 3 (𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) → 𝑅 ∈ (𝐵m 𝐶))
2417, 23opelxpd 5627 . 2 (𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) → ⟨𝐷, 𝑅⟩ ∈ ((𝐴m 𝐶) × (𝐵m 𝐶)))
25 xp1st 7863 . . . . . . 7 (𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶)) → (1st𝑦) ∈ (𝐴m 𝐶))
265, 8elmap 8659 . . . . . . 7 ((1st𝑦) ∈ (𝐴m 𝐶) ↔ (1st𝑦):𝐶𝐴)
2725, 26sylib 217 . . . . . 6 (𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶)) → (1st𝑦):𝐶𝐴)
2827ffvelrnda 6961 . . . . 5 ((𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶)) ∧ 𝑧𝐶) → ((1st𝑦)‘𝑧) ∈ 𝐴)
29 xp2nd 7864 . . . . . . 7 (𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶)) → (2nd𝑦) ∈ (𝐵m 𝐶))
306, 8elmap 8659 . . . . . . 7 ((2nd𝑦) ∈ (𝐵m 𝐶) ↔ (2nd𝑦):𝐶𝐵)
3129, 30sylib 217 . . . . . 6 (𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶)) → (2nd𝑦):𝐶𝐵)
3231ffvelrnda 6961 . . . . 5 ((𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶)) ∧ 𝑧𝐶) → ((2nd𝑦)‘𝑧) ∈ 𝐵)
3328, 32opelxpd 5627 . . . 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 6988 . . 3 (𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶)) → 𝑆:𝐶⟶(𝐴 × 𝐵))
367, 8elmap 8659 . . 3 (𝑆 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ↔ 𝑆:𝐶⟶(𝐴 × 𝐵))
3735, 36sylibr 233 . 2 (𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶)) → 𝑆 ∈ ((𝐴 × 𝐵) ↑m 𝐶))
38 1st2nd2 7870 . . . . 5 (𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶)) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
3938ad2antlr 724 . . . 4 (((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑥 = 𝑆) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
4027feqmptd 6837 . . . . . . 7 (𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶)) → (1st𝑦) = (𝑧𝐶 ↦ ((1st𝑦)‘𝑧)))
4140ad2antlr 724 . . . . . 6 (((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑥 = 𝑆) → (1st𝑦) = (𝑧𝐶 ↦ ((1st𝑦)‘𝑧)))
42 simplr 766 . . . . . . . . . . . 12 ((((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑥 = 𝑆) ∧ 𝑧𝐶) → 𝑥 = 𝑆)
4342fveq1d 6776 . . . . . . . . . . 11 ((((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑥 = 𝑆) ∧ 𝑧𝐶) → (𝑥𝑧) = (𝑆𝑧))
44 opex 5379 . . . . . . . . . . . . 13 ⟨((1st𝑦)‘𝑧), ((2nd𝑦)‘𝑧)⟩ ∈ V
4534fvmpt2 6886 . . . . . . . . . . . . 13 ((𝑧𝐶 ∧ ⟨((1st𝑦)‘𝑧), ((2nd𝑦)‘𝑧)⟩ ∈ V) → (𝑆𝑧) = ⟨((1st𝑦)‘𝑧), ((2nd𝑦)‘𝑧)⟩)
4644, 45mpan2 688 . . . . . . . . . . . 12 (𝑧𝐶 → (𝑆𝑧) = ⟨((1st𝑦)‘𝑧), ((2nd𝑦)‘𝑧)⟩)
4746adantl 482 . . . . . . . . . . 11 ((((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑥 = 𝑆) ∧ 𝑧𝐶) → (𝑆𝑧) = ⟨((1st𝑦)‘𝑧), ((2nd𝑦)‘𝑧)⟩)
4843, 47eqtrd 2778 . . . . . . . . . 10 ((((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑥 = 𝑆) ∧ 𝑧𝐶) → (𝑥𝑧) = ⟨((1st𝑦)‘𝑧), ((2nd𝑦)‘𝑧)⟩)
4948fveq2d 6778 . . . . . . . . 9 ((((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑥 = 𝑆) ∧ 𝑧𝐶) → (1st ‘(𝑥𝑧)) = (1st ‘⟨((1st𝑦)‘𝑧), ((2nd𝑦)‘𝑧)⟩))
50 fvex 6787 . . . . . . . . . 10 ((1st𝑦)‘𝑧) ∈ V
51 fvex 6787 . . . . . . . . . 10 ((2nd𝑦)‘𝑧) ∈ V
5250, 51op1st 7839 . . . . . . . . 9 (1st ‘⟨((1st𝑦)‘𝑧), ((2nd𝑦)‘𝑧)⟩) = ((1st𝑦)‘𝑧)
5349, 52eqtrdi 2794 . . . . . . . 8 ((((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑥 = 𝑆) ∧ 𝑧𝐶) → (1st ‘(𝑥𝑧)) = ((1st𝑦)‘𝑧))
5453mpteq2dva 5174 . . . . . . 7 (((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑥 = 𝑆) → (𝑧𝐶 ↦ (1st ‘(𝑥𝑧))) = (𝑧𝐶 ↦ ((1st𝑦)‘𝑧)))
5514, 54eqtrid 2790 . . . . . 6 (((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑥 = 𝑆) → 𝐷 = (𝑧𝐶 ↦ ((1st𝑦)‘𝑧)))
5641, 55eqtr4d 2781 . . . . 5 (((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑥 = 𝑆) → (1st𝑦) = 𝐷)
5731feqmptd 6837 . . . . . . 7 (𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶)) → (2nd𝑦) = (𝑧𝐶 ↦ ((2nd𝑦)‘𝑧)))
5857ad2antlr 724 . . . . . 6 (((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑥 = 𝑆) → (2nd𝑦) = (𝑧𝐶 ↦ ((2nd𝑦)‘𝑧)))
5948fveq2d 6778 . . . . . . . . 9 ((((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑥 = 𝑆) ∧ 𝑧𝐶) → (2nd ‘(𝑥𝑧)) = (2nd ‘⟨((1st𝑦)‘𝑧), ((2nd𝑦)‘𝑧)⟩))
6050, 51op2nd 7840 . . . . . . . . 9 (2nd ‘⟨((1st𝑦)‘𝑧), ((2nd𝑦)‘𝑧)⟩) = ((2nd𝑦)‘𝑧)
6159, 60eqtrdi 2794 . . . . . . . 8 ((((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑥 = 𝑆) ∧ 𝑧𝐶) → (2nd ‘(𝑥𝑧)) = ((2nd𝑦)‘𝑧))
6261mpteq2dva 5174 . . . . . . 7 (((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑥 = 𝑆) → (𝑧𝐶 ↦ (2nd ‘(𝑥𝑧))) = (𝑧𝐶 ↦ ((2nd𝑦)‘𝑧)))
6320, 62eqtrid 2790 . . . . . 6 (((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑥 = 𝑆) → 𝑅 = (𝑧𝐶 ↦ ((2nd𝑦)‘𝑧)))
6458, 63eqtr4d 2781 . . . . 5 (((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑥 = 𝑆) → (2nd𝑦) = 𝑅)
6556, 64opeq12d 4812 . . . 4 (((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑥 = 𝑆) → ⟨(1st𝑦), (2nd𝑦)⟩ = ⟨𝐷, 𝑅⟩)
6639, 65eqtrd 2778 . . 3 (((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑥 = 𝑆) → 𝑦 = ⟨𝐷, 𝑅⟩)
67 simpll 764 . . . . . 6 (((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → 𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶))
6867, 9sylib 217 . . . . 5 (((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → 𝑥:𝐶⟶(𝐴 × 𝐵))
6968feqmptd 6837 . . . 4 (((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → 𝑥 = (𝑧𝐶 ↦ (𝑥𝑧)))
70 simpr 485 . . . . . . . . . . . 12 (((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → 𝑦 = ⟨𝐷, 𝑅⟩)
7170fveq2d 6778 . . . . . . . . . . 11 (((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → (1st𝑦) = (1st ‘⟨𝐷, 𝑅⟩))
7217ad2antrr 723 . . . . . . . . . . . 12 (((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → 𝐷 ∈ (𝐴m 𝐶))
7323ad2antrr 723 . . . . . . . . . . . 12 (((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → 𝑅 ∈ (𝐵m 𝐶))
74 op1stg 7843 . . . . . . . . . . . 12 ((𝐷 ∈ (𝐴m 𝐶) ∧ 𝑅 ∈ (𝐵m 𝐶)) → (1st ‘⟨𝐷, 𝑅⟩) = 𝐷)
7572, 73, 74syl2anc 584 . . . . . . . . . . 11 (((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → (1st ‘⟨𝐷, 𝑅⟩) = 𝐷)
7671, 75eqtrd 2778 . . . . . . . . . 10 (((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → (1st𝑦) = 𝐷)
7776fveq1d 6776 . . . . . . . . 9 (((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → ((1st𝑦)‘𝑧) = (𝐷𝑧))
78 fvex 6787 . . . . . . . . . 10 (1st ‘(𝑥𝑧)) ∈ V
7914fvmpt2 6886 . . . . . . . . . 10 ((𝑧𝐶 ∧ (1st ‘(𝑥𝑧)) ∈ V) → (𝐷𝑧) = (1st ‘(𝑥𝑧)))
8078, 79mpan2 688 . . . . . . . . 9 (𝑧𝐶 → (𝐷𝑧) = (1st ‘(𝑥𝑧)))
8177, 80sylan9eq 2798 . . . . . . . 8 ((((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) ∧ 𝑧𝐶) → ((1st𝑦)‘𝑧) = (1st ‘(𝑥𝑧)))
8270fveq2d 6778 . . . . . . . . . . 11 (((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → (2nd𝑦) = (2nd ‘⟨𝐷, 𝑅⟩))
83 op2ndg 7844 . . . . . . . . . . . 12 ((𝐷 ∈ (𝐴m 𝐶) ∧ 𝑅 ∈ (𝐵m 𝐶)) → (2nd ‘⟨𝐷, 𝑅⟩) = 𝑅)
8472, 73, 83syl2anc 584 . . . . . . . . . . 11 (((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → (2nd ‘⟨𝐷, 𝑅⟩) = 𝑅)
8582, 84eqtrd 2778 . . . . . . . . . 10 (((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → (2nd𝑦) = 𝑅)
8685fveq1d 6776 . . . . . . . . 9 (((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → ((2nd𝑦)‘𝑧) = (𝑅𝑧))
87 fvex 6787 . . . . . . . . . 10 (2nd ‘(𝑥𝑧)) ∈ V
8820fvmpt2 6886 . . . . . . . . . 10 ((𝑧𝐶 ∧ (2nd ‘(𝑥𝑧)) ∈ V) → (𝑅𝑧) = (2nd ‘(𝑥𝑧)))
8987, 88mpan2 688 . . . . . . . . 9 (𝑧𝐶 → (𝑅𝑧) = (2nd ‘(𝑥𝑧)))
9086, 89sylan9eq 2798 . . . . . . . 8 ((((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) ∧ 𝑧𝐶) → ((2nd𝑦)‘𝑧) = (2nd ‘(𝑥𝑧)))
9181, 90opeq12d 4812 . . . . . . 7 ((((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) ∧ 𝑧𝐶) → ⟨((1st𝑦)‘𝑧), ((2nd𝑦)‘𝑧)⟩ = ⟨(1st ‘(𝑥𝑧)), (2nd ‘(𝑥𝑧))⟩)
9268ffvelrnda 6961 . . . . . . . 8 ((((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) ∧ 𝑧𝐶) → (𝑥𝑧) ∈ (𝐴 × 𝐵))
93 1st2nd2 7870 . . . . . . . 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 2781 . . . . . 6 ((((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) ∧ 𝑧𝐶) → ⟨((1st𝑦)‘𝑧), ((2nd𝑦)‘𝑧)⟩ = (𝑥𝑧))
9695mpteq2dva 5174 . . . . 5 (((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → (𝑧𝐶 ↦ ⟨((1st𝑦)‘𝑧), ((2nd𝑦)‘𝑧)⟩) = (𝑧𝐶 ↦ (𝑥𝑧)))
9734, 96eqtrid 2790 . . . 4 (((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → 𝑆 = (𝑧𝐶 ↦ (𝑥𝑧)))
9869, 97eqtr4d 2781 . . 3 (((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → 𝑥 = 𝑆)
9966, 98impbida 798 . 2 ((𝑥 ∈ ((𝐴 × 𝐵) ↑m 𝐶) ∧ 𝑦 ∈ ((𝐴m 𝐶) × (𝐵m 𝐶))) → (𝑥 = 𝑆𝑦 = ⟨𝐷, 𝑅⟩))
1001, 4, 24, 37, 99en3i 8779 1 ((𝐴 × 𝐵) ↑m 𝐶) ≈ ((𝐴m 𝐶) × (𝐵m 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1539  wcel 2106  Vcvv 3432  cop 4567   class class class wbr 5074  cmpt 5157   × cxp 5587  wf 6429  cfv 6433  (class class class)co 7275  1st c1st 7829  2nd c2nd 7830  m cmap 8615  cen 8730
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-map 8617  df-en 8734
This theorem is referenced by:  xpmapen  8932
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