ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  elxp6 Structured version   GIF version

Theorem elxp6 5738
Description: Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 4751. (Contributed by NM, 9-Oct-2004.)
Assertion
Ref Expression
elxp6 (A (B × 𝐶) ↔ (A = ⟨(1stA), (2ndA)⟩ ((1stA) B (2ndA) 𝐶)))

Proof of Theorem elxp6
StepHypRef Expression
1 elex 2560 . 2 (A (B × 𝐶) → A V)
2 opexg 3955 . . . 4 (((1stA) B (2ndA) 𝐶) → ⟨(1stA), (2ndA)⟩ V)
32adantl 262 . . 3 ((A = ⟨(1stA), (2ndA)⟩ ((1stA) B (2ndA) 𝐶)) → ⟨(1stA), (2ndA)⟩ V)
4 eleq1 2097 . . . 4 (A = ⟨(1stA), (2ndA)⟩ → (A V ↔ ⟨(1stA), (2ndA)⟩ V))
54adantr 261 . . 3 ((A = ⟨(1stA), (2ndA)⟩ ((1stA) B (2ndA) 𝐶)) → (A V ↔ ⟨(1stA), (2ndA)⟩ V))
63, 5mpbird 156 . 2 ((A = ⟨(1stA), (2ndA)⟩ ((1stA) B (2ndA) 𝐶)) → A V)
7 1stvalg 5711 . . . . . 6 (A V → (1stA) = dom {A})
8 2ndvalg 5712 . . . . . 6 (A V → (2ndA) = ran {A})
97, 8opeq12d 3548 . . . . 5 (A V → ⟨(1stA), (2ndA)⟩ = ⟨ dom {A}, ran {A}⟩)
109eqeq2d 2048 . . . 4 (A V → (A = ⟨(1stA), (2ndA)⟩ ↔ A = ⟨ dom {A}, ran {A}⟩))
117eleq1d 2103 . . . . 5 (A V → ((1stA) B dom {A} B))
128eleq1d 2103 . . . . 5 (A V → ((2ndA) 𝐶 ran {A} 𝐶))
1311, 12anbi12d 442 . . . 4 (A V → (((1stA) B (2ndA) 𝐶) ↔ ( dom {A} B ran {A} 𝐶)))
1410, 13anbi12d 442 . . 3 (A V → ((A = ⟨(1stA), (2ndA)⟩ ((1stA) B (2ndA) 𝐶)) ↔ (A = ⟨ dom {A}, ran {A}⟩ ( dom {A} B ran {A} 𝐶))))
15 elxp4 4751 . . 3 (A (B × 𝐶) ↔ (A = ⟨ dom {A}, ran {A}⟩ ( dom {A} B ran {A} 𝐶)))
1614, 15syl6rbbr 188 . 2 (A V → (A (B × 𝐶) ↔ (A = ⟨(1stA), (2ndA)⟩ ((1stA) B (2ndA) 𝐶))))
171, 6, 16pm5.21nii 619 1 (A (B × 𝐶) ↔ (A = ⟨(1stA), (2ndA)⟩ ((1stA) B (2ndA) 𝐶)))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   = wceq 1242   wcel 1390  Vcvv 2551  {csn 3367  cop 3370   cuni 3571   × cxp 4286  dom cdm 4288  ran crn 4289  cfv 4845  1st c1st 5707  2nd c2nd 5708
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fv 4853  df-1st 5709  df-2nd 5710
This theorem is referenced by:  elxp7  5739  eqopi  5740  1st2nd2  5743
  Copyright terms: Public domain W3C validator