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Theorem eqopi 5826
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 4474 . . 3 (𝑉 × 𝑊) ⊆ (V × V)
21sseli 2969 . 2 (𝐴 ∈ (𝑉 × 𝑊) → 𝐴 ∈ (V × V))
3 elxp6 5824 . . . 4 (𝐴 ∈ (V × V) ↔ (𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩ ∧ ((1st𝐴) ∈ V ∧ (2nd𝐴) ∈ V)))
43simplbi 263 . . 3 (𝐴 ∈ (V × V) → 𝐴 = ⟨(1st𝐴), (2nd𝐴)⟩)
5 opeq12 3579 . . 3 (((1st𝐴) = 𝐵 ∧ (2nd𝐴) = 𝐶) → ⟨(1st𝐴), (2nd𝐴)⟩ = ⟨𝐵, 𝐶⟩)
64, 5sylan9eq 2108 . 2 ((𝐴 ∈ (V × V) ∧ ((1st𝐴) = 𝐵 ∧ (2nd𝐴) = 𝐶)) → 𝐴 = ⟨𝐵, 𝐶⟩)
72, 6sylan 271 1 ((𝐴 ∈ (𝑉 × 𝑊) ∧ ((1st𝐴) = 𝐵 ∧ (2nd𝐴) = 𝐶)) → 𝐴 = ⟨𝐵, 𝐶⟩)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1259  wcel 1409  Vcvv 2574  cop 3406   × cxp 4371  cfv 4930  1st c1st 5793  2nd c2nd 5794
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2788  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-br 3793  df-opab 3847  df-mpt 3848  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-iota 4895  df-fun 4932  df-fv 4938  df-1st 5795  df-2nd 5796
This theorem is referenced by:  op1steq  5833  dfoprab3  5845  1stconst  5870  2ndconst  5871  cnvoprab  5883
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