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Theorem xpmapenlem 8079
 Description: Lemma for xpmapen 8080. (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 6638 . 2 ((𝐴 × 𝐵) ↑𝑚 𝐶) ∈ V
2 ovex 6638 . . 3 (𝐴𝑚 𝐶) ∈ V
3 ovex 6638 . . 3 (𝐵𝑚 𝐶) ∈ V
42, 3xpex 6922 . 2 ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶)) ∈ V
5 xpmapen.1 . . . . . . . . 9 𝐴 ∈ V
6 xpmapen.2 . . . . . . . . 9 𝐵 ∈ V
75, 6xpex 6922 . . . . . . . 8 (𝐴 × 𝐵) ∈ V
8 xpmapen.3 . . . . . . . 8 𝐶 ∈ V
97, 8elmap 7838 . . . . . . 7 (𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ↔ 𝑥:𝐶⟶(𝐴 × 𝐵))
10 ffvelrn 6318 . . . . . . 7 ((𝑥:𝐶⟶(𝐴 × 𝐵) ∧ 𝑧𝐶) → (𝑥𝑧) ∈ (𝐴 × 𝐵))
119, 10sylanb 489 . . . . . 6 ((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑧𝐶) → (𝑥𝑧) ∈ (𝐴 × 𝐵))
12 xp1st 7150 . . . . . 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 6346 . . . 4 (𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) → 𝐷:𝐶𝐴)
165, 8elmap 7838 . . . 4 (𝐷 ∈ (𝐴𝑚 𝐶) ↔ 𝐷:𝐶𝐴)
1715, 16sylibr 224 . . 3 (𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) → 𝐷 ∈ (𝐴𝑚 𝐶))
18 xp2nd 7151 . . . . . 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 6346 . . . 4 (𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) → 𝑅:𝐶𝐵)
226, 8elmap 7838 . . . 4 (𝑅 ∈ (𝐵𝑚 𝐶) ↔ 𝑅:𝐶𝐵)
2321, 22sylibr 224 . . 3 (𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) → 𝑅 ∈ (𝐵𝑚 𝐶))
24 opelxpi 5113 . . 3 ((𝐷 ∈ (𝐴𝑚 𝐶) ∧ 𝑅 ∈ (𝐵𝑚 𝐶)) → ⟨𝐷, 𝑅⟩ ∈ ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶)))
2517, 23, 24syl2anc 692 . 2 (𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) → ⟨𝐷, 𝑅⟩ ∈ ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶)))
26 xp1st 7150 . . . . . . 7 (𝑦 ∈ ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶)) → (1st𝑦) ∈ (𝐴𝑚 𝐶))
275, 8elmap 7838 . . . . . . 7 ((1st𝑦) ∈ (𝐴𝑚 𝐶) ↔ (1st𝑦):𝐶𝐴)
2826, 27sylib 208 . . . . . 6 (𝑦 ∈ ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶)) → (1st𝑦):𝐶𝐴)
2928ffvelrnda 6320 . . . . 5 ((𝑦 ∈ ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶)) ∧ 𝑧𝐶) → ((1st𝑦)‘𝑧) ∈ 𝐴)
30 xp2nd 7151 . . . . . . 7 (𝑦 ∈ ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶)) → (2nd𝑦) ∈ (𝐵𝑚 𝐶))
316, 8elmap 7838 . . . . . . 7 ((2nd𝑦) ∈ (𝐵𝑚 𝐶) ↔ (2nd𝑦):𝐶𝐵)
3230, 31sylib 208 . . . . . 6 (𝑦 ∈ ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶)) → (2nd𝑦):𝐶𝐵)
3332ffvelrnda 6320 . . . . 5 ((𝑦 ∈ ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶)) ∧ 𝑧𝐶) → ((2nd𝑦)‘𝑧) ∈ 𝐵)
34 opelxpi 5113 . . . . 5 ((((1st𝑦)‘𝑧) ∈ 𝐴 ∧ ((2nd𝑦)‘𝑧) ∈ 𝐵) → ⟨((1st𝑦)‘𝑧), ((2nd𝑦)‘𝑧)⟩ ∈ (𝐴 × 𝐵))
3529, 33, 34syl2anc 692 . . . 4 ((𝑦 ∈ ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶)) ∧ 𝑧𝐶) → ⟨((1st𝑦)‘𝑧), ((2nd𝑦)‘𝑧)⟩ ∈ (𝐴 × 𝐵))
36 xpmapenlem.6 . . . 4 𝑆 = (𝑧𝐶 ↦ ⟨((1st𝑦)‘𝑧), ((2nd𝑦)‘𝑧)⟩)
3735, 36fmptd 6346 . . 3 (𝑦 ∈ ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶)) → 𝑆:𝐶⟶(𝐴 × 𝐵))
387, 8elmap 7838 . . 3 (𝑆 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ↔ 𝑆:𝐶⟶(𝐴 × 𝐵))
3937, 38sylibr 224 . 2 (𝑦 ∈ ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶)) → 𝑆 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶))
40 1st2nd2 7157 . . . . 5 (𝑦 ∈ ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶)) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
4140ad2antlr 762 . . . 4 (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶))) ∧ 𝑥 = 𝑆) → 𝑦 = ⟨(1st𝑦), (2nd𝑦)⟩)
4228feqmptd 6211 . . . . . . 7 (𝑦 ∈ ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶)) → (1st𝑦) = (𝑧𝐶 ↦ ((1st𝑦)‘𝑧)))
4342ad2antlr 762 . . . . . 6 (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶))) ∧ 𝑥 = 𝑆) → (1st𝑦) = (𝑧𝐶 ↦ ((1st𝑦)‘𝑧)))
44 simplr 791 . . . . . . . . . . . 12 ((((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶))) ∧ 𝑥 = 𝑆) ∧ 𝑧𝐶) → 𝑥 = 𝑆)
4544fveq1d 6155 . . . . . . . . . . 11 ((((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶))) ∧ 𝑥 = 𝑆) ∧ 𝑧𝐶) → (𝑥𝑧) = (𝑆𝑧))
46 opex 4898 . . . . . . . . . . . . 13 ⟨((1st𝑦)‘𝑧), ((2nd𝑦)‘𝑧)⟩ ∈ V
4736fvmpt2 6253 . . . . . . . . . . . . 13 ((𝑧𝐶 ∧ ⟨((1st𝑦)‘𝑧), ((2nd𝑦)‘𝑧)⟩ ∈ V) → (𝑆𝑧) = ⟨((1st𝑦)‘𝑧), ((2nd𝑦)‘𝑧)⟩)
4846, 47mpan2 706 . . . . . . . . . . . 12 (𝑧𝐶 → (𝑆𝑧) = ⟨((1st𝑦)‘𝑧), ((2nd𝑦)‘𝑧)⟩)
4948adantl 482 . . . . . . . . . . 11 ((((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶))) ∧ 𝑥 = 𝑆) ∧ 𝑧𝐶) → (𝑆𝑧) = ⟨((1st𝑦)‘𝑧), ((2nd𝑦)‘𝑧)⟩)
5045, 49eqtrd 2655 . . . . . . . . . 10 ((((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶))) ∧ 𝑥 = 𝑆) ∧ 𝑧𝐶) → (𝑥𝑧) = ⟨((1st𝑦)‘𝑧), ((2nd𝑦)‘𝑧)⟩)
5150fveq2d 6157 . . . . . . . . 9 ((((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶))) ∧ 𝑥 = 𝑆) ∧ 𝑧𝐶) → (1st ‘(𝑥𝑧)) = (1st ‘⟨((1st𝑦)‘𝑧), ((2nd𝑦)‘𝑧)⟩))
52 fvex 6163 . . . . . . . . . 10 ((1st𝑦)‘𝑧) ∈ V
53 fvex 6163 . . . . . . . . . 10 ((2nd𝑦)‘𝑧) ∈ V
5452, 53op1st 7128 . . . . . . . . 9 (1st ‘⟨((1st𝑦)‘𝑧), ((2nd𝑦)‘𝑧)⟩) = ((1st𝑦)‘𝑧)
5551, 54syl6eq 2671 . . . . . . . 8 ((((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶))) ∧ 𝑥 = 𝑆) ∧ 𝑧𝐶) → (1st ‘(𝑥𝑧)) = ((1st𝑦)‘𝑧))
5655mpteq2dva 4709 . . . . . . 7 (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶))) ∧ 𝑥 = 𝑆) → (𝑧𝐶 ↦ (1st ‘(𝑥𝑧))) = (𝑧𝐶 ↦ ((1st𝑦)‘𝑧)))
5714, 56syl5eq 2667 . . . . . 6 (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶))) ∧ 𝑥 = 𝑆) → 𝐷 = (𝑧𝐶 ↦ ((1st𝑦)‘𝑧)))
5843, 57eqtr4d 2658 . . . . 5 (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶))) ∧ 𝑥 = 𝑆) → (1st𝑦) = 𝐷)
5932feqmptd 6211 . . . . . . 7 (𝑦 ∈ ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶)) → (2nd𝑦) = (𝑧𝐶 ↦ ((2nd𝑦)‘𝑧)))
6059ad2antlr 762 . . . . . 6 (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶))) ∧ 𝑥 = 𝑆) → (2nd𝑦) = (𝑧𝐶 ↦ ((2nd𝑦)‘𝑧)))
6150fveq2d 6157 . . . . . . . . 9 ((((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶))) ∧ 𝑥 = 𝑆) ∧ 𝑧𝐶) → (2nd ‘(𝑥𝑧)) = (2nd ‘⟨((1st𝑦)‘𝑧), ((2nd𝑦)‘𝑧)⟩))
6252, 53op2nd 7129 . . . . . . . . 9 (2nd ‘⟨((1st𝑦)‘𝑧), ((2nd𝑦)‘𝑧)⟩) = ((2nd𝑦)‘𝑧)
6361, 62syl6eq 2671 . . . . . . . 8 ((((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶))) ∧ 𝑥 = 𝑆) ∧ 𝑧𝐶) → (2nd ‘(𝑥𝑧)) = ((2nd𝑦)‘𝑧))
6463mpteq2dva 4709 . . . . . . 7 (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶))) ∧ 𝑥 = 𝑆) → (𝑧𝐶 ↦ (2nd ‘(𝑥𝑧))) = (𝑧𝐶 ↦ ((2nd𝑦)‘𝑧)))
6520, 64syl5eq 2667 . . . . . 6 (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶))) ∧ 𝑥 = 𝑆) → 𝑅 = (𝑧𝐶 ↦ ((2nd𝑦)‘𝑧)))
6660, 65eqtr4d 2658 . . . . 5 (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶))) ∧ 𝑥 = 𝑆) → (2nd𝑦) = 𝑅)
6758, 66opeq12d 4383 . . . 4 (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶))) ∧ 𝑥 = 𝑆) → ⟨(1st𝑦), (2nd𝑦)⟩ = ⟨𝐷, 𝑅⟩)
6841, 67eqtrd 2655 . . 3 (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶))) ∧ 𝑥 = 𝑆) → 𝑦 = ⟨𝐷, 𝑅⟩)
69 simpll 789 . . . . . 6 (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → 𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶))
7069, 9sylib 208 . . . . 5 (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → 𝑥:𝐶⟶(𝐴 × 𝐵))
7170feqmptd 6211 . . . 4 (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → 𝑥 = (𝑧𝐶 ↦ (𝑥𝑧)))
72 simpr 477 . . . . . . . . . . . 12 (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → 𝑦 = ⟨𝐷, 𝑅⟩)
7372fveq2d 6157 . . . . . . . . . . 11 (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → (1st𝑦) = (1st ‘⟨𝐷, 𝑅⟩))
7417ad2antrr 761 . . . . . . . . . . . 12 (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → 𝐷 ∈ (𝐴𝑚 𝐶))
7523ad2antrr 761 . . . . . . . . . . . 12 (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → 𝑅 ∈ (𝐵𝑚 𝐶))
76 op1stg 7132 . . . . . . . . . . . 12 ((𝐷 ∈ (𝐴𝑚 𝐶) ∧ 𝑅 ∈ (𝐵𝑚 𝐶)) → (1st ‘⟨𝐷, 𝑅⟩) = 𝐷)
7774, 75, 76syl2anc 692 . . . . . . . . . . 11 (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → (1st ‘⟨𝐷, 𝑅⟩) = 𝐷)
7873, 77eqtrd 2655 . . . . . . . . . 10 (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → (1st𝑦) = 𝐷)
7978fveq1d 6155 . . . . . . . . 9 (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → ((1st𝑦)‘𝑧) = (𝐷𝑧))
80 fvex 6163 . . . . . . . . . 10 (1st ‘(𝑥𝑧)) ∈ V
8114fvmpt2 6253 . . . . . . . . . 10 ((𝑧𝐶 ∧ (1st ‘(𝑥𝑧)) ∈ V) → (𝐷𝑧) = (1st ‘(𝑥𝑧)))
8280, 81mpan2 706 . . . . . . . . 9 (𝑧𝐶 → (𝐷𝑧) = (1st ‘(𝑥𝑧)))
8379, 82sylan9eq 2675 . . . . . . . 8 ((((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) ∧ 𝑧𝐶) → ((1st𝑦)‘𝑧) = (1st ‘(𝑥𝑧)))
8472fveq2d 6157 . . . . . . . . . . 11 (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → (2nd𝑦) = (2nd ‘⟨𝐷, 𝑅⟩))
85 op2ndg 7133 . . . . . . . . . . . 12 ((𝐷 ∈ (𝐴𝑚 𝐶) ∧ 𝑅 ∈ (𝐵𝑚 𝐶)) → (2nd ‘⟨𝐷, 𝑅⟩) = 𝑅)
8674, 75, 85syl2anc 692 . . . . . . . . . . 11 (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → (2nd ‘⟨𝐷, 𝑅⟩) = 𝑅)
8784, 86eqtrd 2655 . . . . . . . . . 10 (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → (2nd𝑦) = 𝑅)
8887fveq1d 6155 . . . . . . . . 9 (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → ((2nd𝑦)‘𝑧) = (𝑅𝑧))
89 fvex 6163 . . . . . . . . . 10 (2nd ‘(𝑥𝑧)) ∈ V
9020fvmpt2 6253 . . . . . . . . . 10 ((𝑧𝐶 ∧ (2nd ‘(𝑥𝑧)) ∈ V) → (𝑅𝑧) = (2nd ‘(𝑥𝑧)))
9189, 90mpan2 706 . . . . . . . . 9 (𝑧𝐶 → (𝑅𝑧) = (2nd ‘(𝑥𝑧)))
9288, 91sylan9eq 2675 . . . . . . . 8 ((((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) ∧ 𝑧𝐶) → ((2nd𝑦)‘𝑧) = (2nd ‘(𝑥𝑧)))
9383, 92opeq12d 4383 . . . . . . 7 ((((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) ∧ 𝑧𝐶) → ⟨((1st𝑦)‘𝑧), ((2nd𝑦)‘𝑧)⟩ = ⟨(1st ‘(𝑥𝑧)), (2nd ‘(𝑥𝑧))⟩)
9470ffvelrnda 6320 . . . . . . . 8 ((((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) ∧ 𝑧𝐶) → (𝑥𝑧) ∈ (𝐴 × 𝐵))
95 1st2nd2 7157 . . . . . . . 8 ((𝑥𝑧) ∈ (𝐴 × 𝐵) → (𝑥𝑧) = ⟨(1st ‘(𝑥𝑧)), (2nd ‘(𝑥𝑧))⟩)
9694, 95syl 17 . . . . . . 7 ((((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) ∧ 𝑧𝐶) → (𝑥𝑧) = ⟨(1st ‘(𝑥𝑧)), (2nd ‘(𝑥𝑧))⟩)
9793, 96eqtr4d 2658 . . . . . 6 ((((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) ∧ 𝑧𝐶) → ⟨((1st𝑦)‘𝑧), ((2nd𝑦)‘𝑧)⟩ = (𝑥𝑧))
9897mpteq2dva 4709 . . . . 5 (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → (𝑧𝐶 ↦ ⟨((1st𝑦)‘𝑧), ((2nd𝑦)‘𝑧)⟩) = (𝑧𝐶 ↦ (𝑥𝑧)))
9936, 98syl5eq 2667 . . . 4 (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → 𝑆 = (𝑧𝐶 ↦ (𝑥𝑧)))
10071, 99eqtr4d 2658 . . 3 (((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶))) ∧ 𝑦 = ⟨𝐷, 𝑅⟩) → 𝑥 = 𝑆)
10168, 100impbida 876 . 2 ((𝑥 ∈ ((𝐴 × 𝐵) ↑𝑚 𝐶) ∧ 𝑦 ∈ ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶))) → (𝑥 = 𝑆𝑦 = ⟨𝐷, 𝑅⟩))
1021, 4, 25, 39, 101en3i 7946 1 ((𝐴 × 𝐵) ↑𝑚 𝐶) ≈ ((𝐴𝑚 𝐶) × (𝐵𝑚 𝐶))
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 384   = wceq 1480   ∈ wcel 1987  Vcvv 3189  ⟨cop 4159   class class class wbr 4618   ↦ cmpt 4678   × cxp 5077  ⟶wf 5848  ‘cfv 5852  (class class class)co 6610  1st c1st 7118  2nd c2nd 7119   ↑𝑚 cmap 7809   ≈ cen 7904 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-ov 6613  df-oprab 6614  df-mpt2 6615  df-1st 7120  df-2nd 7121  df-map 7811  df-en 7908 This theorem is referenced by:  xpmapen  8080
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