Theorem eqopi 6396

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) ∧ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶)) → 𝐴 = ⟨𝐵, 𝐶⟩)

Proof of Theorem eqopi

Step	Hyp	Ref	Expression
1		xpss 4878	. . 3 ⊢ (𝑉 × 𝑊) ⊆ (V × V)
2	1	sseli 3244	. 2 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → 𝐴 ∈ (V × V))
3		elxp6 6393	. . . 4 ⊢ (𝐴 ∈ (V × V) ↔ (𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ ∧ ((1st ‘𝐴) ∈ V ∧ (2nd ‘𝐴) ∈ V)))
4	3	simplbi 274	. . 3 ⊢ (𝐴 ∈ (V × V) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
5		opeq12 3901	. . 3 ⊢ (((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶) → ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ = ⟨𝐵, 𝐶⟩)
6	4, 5	sylan9eq 2291	. 2 ⊢ ((𝐴 ∈ (V × V) ∧ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶)) → 𝐴 = ⟨𝐵, 𝐶⟩)
7	2, 6	sylan 283	1 ⊢ ((𝐴 ∈ (𝑉 × 𝑊) ∧ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶)) → 𝐴 = ⟨𝐵, 𝐶⟩)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1402   ∈ wcel 2209  Vcvv 2821  ⟨cop 3708   × cxp 4767  ‘cfv 5372  1^st c1st 6362  2^nd c2nd 6363

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-iota 5332  df-fun 5374  df-fv 5380  df-1st 6364  df-2nd 6365

This theorem is referenced by:  op1steq  6403  dfoprab3  6415  1stconst  6447  2ndconst  6448  cnvoprab  6460  upxp  15296