MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elxp6 Structured version   Visualization version   GIF version

Theorem elxp6 7244
Description: Membership in a Cartesian product. This version requires no quantifiers or dummy variables. See also elxp4 7152. (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 7152 . 2 (𝐴 ∈ (𝐵 × 𝐶) ↔ (𝐴 = ⟨ dom {𝐴}, ran {𝐴}⟩ ∧ ( dom {𝐴} ∈ 𝐵 ran {𝐴} ∈ 𝐶)))
2 1stval 7212 . . . . 5 (1st𝐴) = dom {𝐴}
3 2ndval 7213 . . . . 5 (2nd𝐴) = ran {𝐴}
42, 3opeq12i 4438 . . . 4 ⟨(1st𝐴), (2nd𝐴)⟩ = ⟨ dom {𝐴}, ran {𝐴}⟩
54eqeq2i 2663 . . 3 (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ↔ 𝐴 = ⟨ dom {𝐴}, ran {𝐴}⟩)
62eleq1i 2721 . . . 4 ((1st𝐴) ∈ 𝐵 dom {𝐴} ∈ 𝐵)
73eleq1i 2721 . . . 4 ((2nd𝐴) ∈ 𝐶 ran {𝐴} ∈ 𝐶)
86, 7anbi12i 733 . . 3 (((1st𝐴) ∈ 𝐵 ∧ (2nd𝐴) ∈ 𝐶) ↔ ( dom {𝐴} ∈ 𝐵 ran {𝐴} ∈ 𝐶))
95, 8anbi12i 733 . 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 383   = wceq 1523  wcel 2030  {csn 4210  cop 4216   cuni 4468   × cxp 5141  dom cdm 5143  ran crn 5144  cfv 5926  1st c1st 7208  2nd c2nd 7209
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fv 5934  df-1st 7210  df-2nd 7211
This theorem is referenced by:  elxp7  7245  eqopi  7246  1st2nd2  7249  r0weon  8873  qredeu  15419  qnumdencl  15494  setsstruct2  15943  tx1cn  21460  tx2cn  21461  txhaus  21498  psmetxrge0  22165  xppreima  29577  ofpreima2  29594  smatrcl  29990  1stmbfm  30450  2ndmbfm  30451  oddpwdcv  30545
  Copyright terms: Public domain W3C validator