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Theorem opeliunxp2 5839
Description: Membership in a union of Cartesian products. (Contributed by Mario Carneiro, 14-Feb-2015.)
Hypothesis
Ref Expression
opeliunxp2.1 (𝑥 = 𝐶𝐵 = 𝐸)
Assertion
Ref Expression
opeliunxp2 (⟨𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝐶𝐴𝐷𝐸))
Distinct variable groups:   𝑥,𝐶   𝑥,𝐷   𝑥,𝐸   𝑥,𝐴
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem opeliunxp2
StepHypRef Expression
1 df-br 5150 . . 3 (𝐶 𝑥𝐴 ({𝑥} × 𝐵)𝐷 ↔ ⟨𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵))
2 relxp 5695 . . . . . 6 Rel ({𝑥} × 𝐵)
32rgenw 3066 . . . . 5 𝑥𝐴 Rel ({𝑥} × 𝐵)
4 reliun 5817 . . . . 5 (Rel 𝑥𝐴 ({𝑥} × 𝐵) ↔ ∀𝑥𝐴 Rel ({𝑥} × 𝐵))
53, 4mpbir 230 . . . 4 Rel 𝑥𝐴 ({𝑥} × 𝐵)
65brrelex1i 5733 . . 3 (𝐶 𝑥𝐴 ({𝑥} × 𝐵)𝐷𝐶 ∈ V)
71, 6sylbir 234 . 2 (⟨𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) → 𝐶 ∈ V)
8 elex 3493 . . 3 (𝐶𝐴𝐶 ∈ V)
98adantr 482 . 2 ((𝐶𝐴𝐷𝐸) → 𝐶 ∈ V)
10 nfiu1 5032 . . . . 5 𝑥 𝑥𝐴 ({𝑥} × 𝐵)
1110nfel2 2922 . . . 4 𝑥𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵)
12 nfv 1918 . . . 4 𝑥(𝐶𝐴𝐷𝐸)
1311, 12nfbi 1907 . . 3 𝑥(⟨𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝐶𝐴𝐷𝐸))
14 opeq1 4874 . . . . 5 (𝑥 = 𝐶 → ⟨𝑥, 𝐷⟩ = ⟨𝐶, 𝐷⟩)
1514eleq1d 2819 . . . 4 (𝑥 = 𝐶 → (⟨𝑥, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ ⟨𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵)))
16 eleq1 2822 . . . . 5 (𝑥 = 𝐶 → (𝑥𝐴𝐶𝐴))
17 opeliunxp2.1 . . . . . 6 (𝑥 = 𝐶𝐵 = 𝐸)
1817eleq2d 2820 . . . . 5 (𝑥 = 𝐶 → (𝐷𝐵𝐷𝐸))
1916, 18anbi12d 632 . . . 4 (𝑥 = 𝐶 → ((𝑥𝐴𝐷𝐵) ↔ (𝐶𝐴𝐷𝐸)))
2015, 19bibi12d 346 . . 3 (𝑥 = 𝐶 → ((⟨𝑥, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝑥𝐴𝐷𝐵)) ↔ (⟨𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝐶𝐴𝐷𝐸))))
21 opeliunxp 5744 . . 3 (⟨𝑥, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝑥𝐴𝐷𝐵))
2213, 20, 21vtoclg1f 3556 . 2 (𝐶 ∈ V → (⟨𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝐶𝐴𝐷𝐸)))
237, 9, 22pm5.21nii 380 1 (⟨𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝐶𝐴𝐷𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3062  Vcvv 3475  {csn 4629  cop 4635   ciun 4998   class class class wbr 5149   × cxp 5675  Rel wrel 5682
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 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-iun 5000  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684
This theorem is referenced by:  mpoxopn0yelv  8198  mpoxopxnop0  8200  eldmcoa  18015  dmdprd  19868  ply1frcl  21837  cnextfres  23573  eldv  25415  perfdvf  25420  eltayl  25872  dfcnv2  31901  cvmliftlem1  34276  filnetlem3  35265
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