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Theorem eqopi 7187
Description: Equality with an ordered pair. (Contributed by NM, 15-Dec-2008.) (Revised by Mario Carneiro, 23-Feb-2014.)
Assertion
Ref Expression
eqopi ((𝐴 ∈ (𝑉 × 𝑊) ∧ ((1st𝐴) = 𝐵 ∧ (2nd𝐴) = 𝐶)) → 𝐴 = ⟨𝐵, 𝐶⟩)

Proof of Theorem eqopi
StepHypRef Expression
1 xpss 5216 . . 3 (𝑉 × 𝑊) ⊆ (V × V)
21sseli 3591 . 2 (𝐴 ∈ (𝑉 × 𝑊) → 𝐴 ∈ (V × V))
3 elxp6 7185 . . . 4 (𝐴 ∈ (V × V) ↔ (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ∧ ((1st𝐴) ∈ V ∧ (2nd𝐴) ∈ V)))
43simplbi 476 . . 3 (𝐴 ∈ (V × V) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
5 opeq12 4395 . . 3 (((1st𝐴) = 𝐵 ∧ (2nd𝐴) = 𝐶) → ⟨(1st𝐴), (2nd𝐴)⟩ = ⟨𝐵, 𝐶⟩)
64, 5sylan9eq 2674 . 2 ((𝐴 ∈ (V × V) ∧ ((1st𝐴) = 𝐵 ∧ (2nd𝐴) = 𝐶)) → 𝐴 = ⟨𝐵, 𝐶⟩)
72, 6sylan 488 1 ((𝐴 ∈ (𝑉 × 𝑊) ∧ ((1st𝐴) = 𝐵 ∧ (2nd𝐴) = 𝐶)) → 𝐴 = ⟨𝐵, 𝐶⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1481  wcel 1988  Vcvv 3195  cop 4174   × cxp 5102  cfv 5876  1st c1st 7151  2nd c2nd 7152
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-iota 5839  df-fun 5878  df-fv 5884  df-1st 7153  df-2nd 7154
This theorem is referenced by:  op1steq  7195  el2xptp0  7197  dfoprab3  7209  1stconst  7250  2ndconst  7251  upxp  21407  cnvoprab  29472  gsummpt2d  29755  sitgaddlemb  30384
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