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Theorem elxp6 7148
Description: Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp4 7060. (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 elxp4 7060 . 2 (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨ dom {𝐴}, ran {𝐴}⟩ ∧ ( dom {𝐴} ∈ 𝐵 ran {𝐴} ∈ 𝐶)))
2 1stval 7118 . . . . 5 (1st𝐴) = dom {𝐴}
3 2ndval 7119 . . . . 5 (2nd𝐴) = ran {𝐴}
42, 3opeq12i 4380 . . . 4 ⟨(1st𝐴), (2nd𝐴)⟩ = ⟨ dom {𝐴}, ran {𝐴}⟩
54eqeq2i 2638 . . 3 (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ↔ 𝐴 = ⟨ dom {𝐴}, ran {𝐴}⟩)
62eleq1i 2695 . . . 4 ((1st𝐴) ∈ 𝐵 dom {𝐴} ∈ 𝐵)
73eleq1i 2695 . . . 4 ((2nd𝐴) ∈ 𝐶 ran {𝐴} ∈ 𝐶)
86, 7anbi12i 732 . . 3 (((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶) ↔ ( dom {𝐴} ∈ 𝐵 ran {𝐴} ∈ 𝐶))
95, 8anbi12i 732 . 2 ((𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)) ↔ (𝐴 = ⟨ dom {𝐴}, ran {𝐴}⟩ ∧ ( dom {𝐴} ∈ 𝐵 ran {𝐴} ∈ 𝐶)))
101, 9bitr4i 267 1 (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ∧ ((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wcel 1992  {csn 4153  cop 4159   cuni 4407   × cxp 5077  dom cdm 5079  ran crn 5080  cfv 5850  1st c1st 7114  2nd c2nd 7115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-iota 5813  df-fun 5852  df-fv 5858  df-1st 7116  df-2nd 7117
This theorem is referenced by:  elxp7  7149  eqopi  7150  1st2nd2  7153  r0weon  8780  qredeu  15291  qnumdencl  15366  setsstruct2  15812  tx1cn  21317  tx2cn  21318  txhaus  21355  psmetxrge0  22023  xppreima  29282  ofpreima2  29300  smatrcl  29636  1stmbfm  30095  2ndmbfm  30096  oddpwdcv  30190
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