Theorem eqop 7846

Description: Two ways to express equality with an ordered pair. (Contributed by NM, 3-Sep-2007.) (Proof shortened by Mario Carneiro, 26-Apr-2015.)

Assertion

Ref	Expression
eqop	⊢ (𝐴 ∈ (𝑉 × 𝑊) → (𝐴 = ⟨𝐵, 𝐶⟩ ↔ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶)))

Proof of Theorem eqop

Step	Hyp	Ref	Expression
1		1st2nd2 7843	. . 3 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
2	1	eqeq1d 2740	. 2 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → (𝐴 = ⟨𝐵, 𝐶⟩ ↔ ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ = ⟨𝐵, 𝐶⟩))
3		fvex 6769	. . 3 ⊢ (1st ‘𝐴) ∈ V
4		fvex 6769	. . 3 ⊢ (2nd ‘𝐴) ∈ V
5	3, 4	opth 5385	. 2 ⊢ (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ = ⟨𝐵, 𝐶⟩ ↔ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶))
6	2, 5	bitrdi 286	1 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → (𝐴 = ⟨𝐵, 𝐶⟩ ↔ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ⟨cop 4564   × cxp 5578  ‘cfv 6418  1^st c1st 7802  2^nd c2nd 7803

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-iota 6376  df-fun 6420  df-fv 6426  df-1st 7804  df-2nd 7805

This theorem is referenced by:  eqop2  7847  op1steq  7848  el2xptp0  7851  lsmhash  19226  txhmeo  22862  ptuncnv  22866  wlkcomp  27900  clwlkcomp  28048  f1od2  30958  esum2dlem  31960  poimirlem22  35726  rngosn3  36009  dvhb1dimN  38927