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Theorem opeliunxp2 5416
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 4805 . . 3 (𝐶 𝑥𝐴 ({𝑥} × 𝐵)𝐷 ↔ ⟨𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵))
2 relxp 5283 . . . . . 6 Rel ({𝑥} × 𝐵)
32rgenw 3062 . . . . 5 𝑥𝐴 Rel ({𝑥} × 𝐵)
4 reliun 5395 . . . . 5 (Rel 𝑥𝐴 ({𝑥} × 𝐵) ↔ ∀𝑥𝐴 Rel ({𝑥} × 𝐵))
53, 4mpbir 221 . . . 4 Rel 𝑥𝐴 ({𝑥} × 𝐵)
65brrelexi 5315 . . 3 (𝐶 𝑥𝐴 ({𝑥} × 𝐵)𝐷𝐶 ∈ V)
71, 6sylbir 225 . 2 (⟨𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) → 𝐶 ∈ V)
8 elex 3352 . . 3 (𝐶𝐴𝐶 ∈ V)
98adantr 472 . 2 ((𝐶𝐴𝐷𝐸) → 𝐶 ∈ V)
10 nfiu1 4702 . . . . 5 𝑥 𝑥𝐴 ({𝑥} × 𝐵)
1110nfel2 2919 . . . 4 𝑥𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵)
12 nfv 1992 . . . 4 𝑥(𝐶𝐴𝐷𝐸)
1311, 12nfbi 1982 . . 3 𝑥(⟨𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝐶𝐴𝐷𝐸))
14 opeq1 4553 . . . . 5 (𝑥 = 𝐶 → ⟨𝑥, 𝐷⟩ = ⟨𝐶, 𝐷⟩)
1514eleq1d 2824 . . . 4 (𝑥 = 𝐶 → (⟨𝑥, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ ⟨𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵)))
16 eleq1 2827 . . . . 5 (𝑥 = 𝐶 → (𝑥𝐴𝐶𝐴))
17 opeliunxp2.1 . . . . . 6 (𝑥 = 𝐶𝐵 = 𝐸)
1817eleq2d 2825 . . . . 5 (𝑥 = 𝐶 → (𝐷𝐵𝐷𝐸))
1916, 18anbi12d 749 . . . 4 (𝑥 = 𝐶 → ((𝑥𝐴𝐷𝐵) ↔ (𝐶𝐴𝐷𝐸)))
2015, 19bibi12d 334 . . 3 (𝑥 = 𝐶 → ((⟨𝑥, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝑥𝐴𝐷𝐵)) ↔ (⟨𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝐶𝐴𝐷𝐸))))
21 opeliunxp 5327 . . 3 (⟨𝑥, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝑥𝐴𝐷𝐵))
2213, 20, 21vtoclg1f 3405 . 2 (𝐶 ∈ V → (⟨𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝐶𝐴𝐷𝐸)))
237, 9, 22pm5.21nii 367 1 (⟨𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝐶𝐴𝐷𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wral 3050  Vcvv 3340  {csn 4321  cop 4327   ciun 4672   class class class wbr 4804   × cxp 5264  Rel wrel 5271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-iun 4674  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273
This theorem is referenced by:  mpt2xopn0yelv  7508  mpt2xopxnop0  7510  eldmcoa  16916  dmdprd  18597  ply1frcl  19885  cnextfres  22074  eldv  23861  perfdvf  23866  eltayl  24313  dfcnv2  29785  cvmliftlem1  31574  filnetlem3  32681
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