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Theorem opelxp 4420
Description: Ordered pair membership in a cross product. (Contributed by NM, 15-Nov-1994.) (Proof shortened by Andrew Salmon, 12-Aug-2011.) (Revised by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
opelxp (⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷) ↔ (𝐴𝐶𝐵𝐷))

Proof of Theorem opelxp
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elxp2 4409 . 2 (⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷) ↔ ∃𝑥𝐶𝑦𝐷𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩)
2 vex 2613 . . . . . . 7 𝑥 ∈ V
3 vex 2613 . . . . . . 7 𝑦 ∈ V
42, 3opth2 4023 . . . . . 6 (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ↔ (𝐴 = 𝑥𝐵 = 𝑦))
5 eleq1 2145 . . . . . . 7 (𝐴 = 𝑥 → (𝐴𝐶𝑥𝐶))
6 eleq1 2145 . . . . . . 7 (𝐵 = 𝑦 → (𝐵𝐷𝑦𝐷))
75, 6bi2anan9 571 . . . . . 6 ((𝐴 = 𝑥𝐵 = 𝑦) → ((𝐴𝐶𝐵𝐷) ↔ (𝑥𝐶𝑦𝐷)))
84, 7sylbi 119 . . . . 5 (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ → ((𝐴𝐶𝐵𝐷) ↔ (𝑥𝐶𝑦𝐷)))
98biimprcd 158 . . . 4 ((𝑥𝐶𝑦𝐷) → (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ → (𝐴𝐶𝐵𝐷)))
109rexlimivv 2487 . . 3 (∃𝑥𝐶𝑦𝐷𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ → (𝐴𝐶𝐵𝐷))
11 eqid 2083 . . . 4 𝐴, 𝐵⟩ = ⟨𝐴, 𝐵
12 opeq1 3590 . . . . . 6 (𝑥 = 𝐴 → ⟨𝑥, 𝑦⟩ = ⟨𝐴, 𝑦⟩)
1312eqeq2d 2094 . . . . 5 (𝑥 = 𝐴 → (⟨𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ↔ ⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝑦⟩))
14 opeq2 3591 . . . . . 6 (𝑦 = 𝐵 → ⟨𝐴, 𝑦⟩ = ⟨𝐴, 𝐵⟩)
1514eqeq2d 2094 . . . . 5 (𝑦 = 𝐵 → (⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝑦⟩ ↔ ⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐵⟩))
1613, 15rspc2ev 2723 . . . 4 ((𝐴𝐶𝐵𝐷 ∧ ⟨𝐴, 𝐵⟩ = ⟨𝐴, 𝐵⟩) → ∃𝑥𝐶𝑦𝐷𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩)
1711, 16mp3an3 1258 . . 3 ((𝐴𝐶𝐵𝐷) → ∃𝑥𝐶𝑦𝐷𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩)
1810, 17impbii 124 . 2 (∃𝑥𝐶𝑦𝐷𝐴, 𝐵⟩ = ⟨𝑥, 𝑦⟩ ↔ (𝐴𝐶𝐵𝐷))
191, 18bitri 182 1 (⟨𝐴, 𝐵⟩ ∈ (𝐶 × 𝐷) ↔ (𝐴𝐶𝐵𝐷))
Colors of variables: wff set class
Syntax hints:  wa 102  wb 103   = wceq 1285  wcel 1434  wrex 2354  cop 3419   × cxp 4389
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-pow 3968  ax-pr 3992
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-v 2612  df-un 2986  df-in 2988  df-ss 2995  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-opab 3860  df-xp 4397
This theorem is referenced by:  brxp  4421  opelxpi  4422  opelxp1  4423  opelxp2  4424  opthprc  4437  elxp3  4440  opeliunxp  4441  optocl  4462  xpiindim  4521  opelres  4665  resiexg  4703  codir  4763  qfto  4764  xpmlem  4794  rnxpid  4805  ssrnres  4813  dfco2  4870  relssdmrn  4891  ressn  4908  opelf  5113  fnovex  5590  oprab4  5627  resoprab  5649  elmpt2cl  5750  fo1stresm  5840  fo2ndresm  5841  dfoprab4  5870  xporderlem  5904  f1od2  5908  brecop  6284  xpdom2  6397  djuss  6564  enq0enq  6753  enq0sym  6754  enq0tr  6756  nqnq0pi  6760  nnnq0lem1  6768  elinp  6796  genipv  6831  prsrlem1  7051  gt0srpr  7057  opelcn  7127  opelreal  7128  elreal2  7131  frecuzrdgrrn  9560  frec2uzrdg  9561  frecuzrdgrcl  9562  frecuzrdgsuc  9566  frecuzrdgrclt  9567  frecuzrdgsuctlem  9575  sqpweven  10778  2sqpwodd  10779  phimullem  10826
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