MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  opeliunxp2 Structured version   Visualization version   GIF version

Theorem opeliunxp2 5464
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 4844 . . 3 (𝐶 𝑥𝐴 ({𝑥} × 𝐵)𝐷 ↔ ⟨𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵))
2 relxp 5330 . . . . . 6 Rel ({𝑥} × 𝐵)
32rgenw 3105 . . . . 5 𝑥𝐴 Rel ({𝑥} × 𝐵)
4 reliun 5443 . . . . 5 (Rel 𝑥𝐴 ({𝑥} × 𝐵) ↔ ∀𝑥𝐴 Rel ({𝑥} × 𝐵))
53, 4mpbir 223 . . . 4 Rel 𝑥𝐴 ({𝑥} × 𝐵)
65brrelex1i 5363 . . 3 (𝐶 𝑥𝐴 ({𝑥} × 𝐵)𝐷𝐶 ∈ V)
71, 6sylbir 227 . 2 (⟨𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) → 𝐶 ∈ V)
8 elex 3400 . . 3 (𝐶𝐴𝐶 ∈ V)
98adantr 473 . 2 ((𝐶𝐴𝐷𝐸) → 𝐶 ∈ V)
10 nfiu1 4740 . . . . 5 𝑥 𝑥𝐴 ({𝑥} × 𝐵)
1110nfel2 2958 . . . 4 𝑥𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵)
12 nfv 2010 . . . 4 𝑥(𝐶𝐴𝐷𝐸)
1311, 12nfbi 2003 . . 3 𝑥(⟨𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝐶𝐴𝐷𝐸))
14 opeq1 4593 . . . . 5 (𝑥 = 𝐶 → ⟨𝑥, 𝐷⟩ = ⟨𝐶, 𝐷⟩)
1514eleq1d 2863 . . . 4 (𝑥 = 𝐶 → (⟨𝑥, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ ⟨𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵)))
16 eleq1 2866 . . . . 5 (𝑥 = 𝐶 → (𝑥𝐴𝐶𝐴))
17 opeliunxp2.1 . . . . . 6 (𝑥 = 𝐶𝐵 = 𝐸)
1817eleq2d 2864 . . . . 5 (𝑥 = 𝐶 → (𝐷𝐵𝐷𝐸))
1916, 18anbi12d 625 . . . 4 (𝑥 = 𝐶 → ((𝑥𝐴𝐷𝐵) ↔ (𝐶𝐴𝐷𝐸)))
2015, 19bibi12d 337 . . 3 (𝑥 = 𝐶 → ((⟨𝑥, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝑥𝐴𝐷𝐵)) ↔ (⟨𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝐶𝐴𝐷𝐸))))
21 opeliunxp 5373 . . 3 (⟨𝑥, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝑥𝐴𝐷𝐵))
2213, 20, 21vtoclg1f 3452 . 2 (𝐶 ∈ V → (⟨𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝐶𝐴𝐷𝐸)))
237, 9, 22pm5.21nii 370 1 (⟨𝐶, 𝐷⟩ ∈ 𝑥𝐴 ({𝑥} × 𝐵) ↔ (𝐶𝐴𝐷𝐸))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 385   = wceq 1653  wcel 2157  wral 3089  Vcvv 3385  {csn 4368  cop 4374   ciun 4710   class class class wbr 4843   × cxp 5310  Rel wrel 5317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-iun 4712  df-br 4844  df-opab 4906  df-xp 5318  df-rel 5319
This theorem is referenced by:  mpt2xopn0yelv  7577  mpt2xopxnop0  7579  eldmcoa  17029  dmdprd  18713  ply1frcl  20005  cnextfres  22201  eldv  24003  perfdvf  24008  eltayl  24455  dfcnv2  29994  cvmliftlem1  31784  filnetlem3  32887
  Copyright terms: Public domain W3C validator