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Theorem elxp6 5715
Description: Membership in a cross product. This version requires no quantifiers or dummy variables. See also elxp4 4731. (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 2539 . 2 (A (B × 𝐶) → A V)
2 opexg 3934 . . . 4 (((1stA) B (2ndA) 𝐶) → ⟨(1stA), (2ndA)⟩ V)
32adantl 262 . . 3 ((A = ⟨(1stA), (2ndA)⟩ ((1stA) B (2ndA) 𝐶)) → ⟨(1stA), (2ndA)⟩ V)
4 eleq1 2078 . . . 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 5688 . . . . . 6 (A V → (1stA) = dom {A})
8 2ndvalg 5689 . . . . . 6 (A V → (2ndA) = ran {A})
97, 8opeq12d 3527 . . . . 5 (A V → ⟨(1stA), (2ndA)⟩ = ⟨ dom {A}, ran {A}⟩)
109eqeq2d 2029 . . . 4 (A V → (A = ⟨(1stA), (2ndA)⟩ ↔ A = ⟨ dom {A}, ran {A}⟩))
117eleq1d 2084 . . . . 5 (A V → ((1stA) B dom {A} B))
128eleq1d 2084 . . . . 5 (A V → ((2ndA) 𝐶 ran {A} 𝐶))
1311, 12anbi12d 445 . . . 4 (A V → (((1stA) B (2ndA) 𝐶) ↔ ( dom {A} B ran {A} 𝐶)))
1410, 13anbi12d 445 . . 3 (A V → ((A = ⟨(1stA), (2ndA)⟩ ((1stA) B (2ndA) 𝐶)) ↔ (A = ⟨ dom {A}, ran {A}⟩ ( dom {A} B ran {A} 𝐶))))
15 elxp4 4731 . . 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 607 1 (A (B × 𝐶) ↔ (A = ⟨(1stA), (2ndA)⟩ ((1stA) B (2ndA) 𝐶)))
Colors of variables: wff set class
Syntax hints:   wa 97  wb 98   = wceq 1226   wcel 1370  Vcvv 2531  {csn 3346  cop 3349   cuni 3550   × cxp 4266  dom cdm 4268  ran crn 4269  cfv 4825  1st c1st 5684  2nd c2nd 5685
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-13 1381  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  ax-un 4116
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-sbc 2738  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-br 3735  df-opab 3789  df-mpt 3790  df-id 4000  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-rn 4279  df-iota 4790  df-fun 4827  df-fv 4833  df-1st 5686  df-2nd 5687
This theorem is referenced by:  elxp7  5716  eqopi  5717  1st2nd2  5720
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