ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  opelxp Structured version   GIF version

Theorem opelxp 4297
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) ↔ (A 𝐶 B 𝐷))

Proof of Theorem opelxp
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elxp2 4286 . 2 (⟨A, B (𝐶 × 𝐷) ↔ x 𝐶 y 𝐷A, B⟩ = ⟨x, y⟩)
2 vex 2534 . . . . . . 7 x V
3 vex 2534 . . . . . . 7 y V
42, 3opth2 3947 . . . . . 6 (⟨A, B⟩ = ⟨x, y⟩ ↔ (A = x B = y))
5 eleq1 2078 . . . . . . 7 (A = x → (A 𝐶x 𝐶))
6 eleq1 2078 . . . . . . 7 (B = y → (B 𝐷y 𝐷))
75, 6bi2anan9 526 . . . . . 6 ((A = x B = y) → ((A 𝐶 B 𝐷) ↔ (x 𝐶 y 𝐷)))
84, 7sylbi 114 . . . . 5 (⟨A, B⟩ = ⟨x, y⟩ → ((A 𝐶 B 𝐷) ↔ (x 𝐶 y 𝐷)))
98biimprcd 149 . . . 4 ((x 𝐶 y 𝐷) → (⟨A, B⟩ = ⟨x, y⟩ → (A 𝐶 B 𝐷)))
109rexlimivv 2412 . . 3 (x 𝐶 y 𝐷A, B⟩ = ⟨x, y⟩ → (A 𝐶 B 𝐷))
11 eqid 2018 . . . 4 A, B⟩ = ⟨A, B
12 opeq1 3519 . . . . . 6 (x = A → ⟨x, y⟩ = ⟨A, y⟩)
1312eqeq2d 2029 . . . . 5 (x = A → (⟨A, B⟩ = ⟨x, y⟩ ↔ ⟨A, B⟩ = ⟨A, y⟩))
14 opeq2 3520 . . . . . 6 (y = B → ⟨A, y⟩ = ⟨A, B⟩)
1514eqeq2d 2029 . . . . 5 (y = B → (⟨A, B⟩ = ⟨A, y⟩ ↔ ⟨A, B⟩ = ⟨A, B⟩))
1613, 15rspc2ev 2637 . . . 4 ((A 𝐶 B 𝐷 A, B⟩ = ⟨A, B⟩) → x 𝐶 y 𝐷A, B⟩ = ⟨x, y⟩)
1711, 16mp3an3 1204 . . 3 ((A 𝐶 B 𝐷) → x 𝐶 y 𝐷A, B⟩ = ⟨x, y⟩)
1810, 17impbii 117 . 2 (x 𝐶 y 𝐷A, B⟩ = ⟨x, y⟩ ↔ (A 𝐶 B 𝐷))
191, 18bitri 173 1 (⟨A, B (𝐶 × 𝐷) ↔ (A 𝐶 B 𝐷))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   = wceq 1226   wcel 1370  wrex 2281  cop 3349   × cxp 4266
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-opab 3789  df-xp 4274
This theorem is referenced by:  brxp  4298  opelxpi  4299  opelxp1  4300  opelxp2  4301  opthprc  4314  elxp3  4317  opeliunxp  4318  optocl  4339  xpiindim  4396  opelres  4540  resiexg  4576  codir  4636  qfto  4637  xpmlem  4667  rnxpid  4678  ssrnres  4686  dfco2  4743  relssdmrn  4764  ressn  4781  opelf  4983  fnovex  5458  oprab4  5494  resoprab  5516  elmpt2cl  5617  fo1stresm  5707  fo2ndresm  5708  dfoprab4  5737  xporderlem  5770  brecop  6103  enq0enq  6280  enq0sym  6281  enq0tr  6283  nqnq0pi  6287  nnnq0lem1  6295  elinp  6322  genipv  6357  prsrlem1  6483  gt0srpr  6489  opelcn  6533  opelreal  6534  elreal2  6536
  Copyright terms: Public domain W3C validator