Theorem eqop 7443

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 7440	. . 3 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
2	1	eqeq1d 2801	. 2 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → (𝐴 = ⟨𝐵, 𝐶⟩ ↔ ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ = ⟨𝐵, 𝐶⟩))
3		fvex 6424	. . 3 ⊢ (1st ‘𝐴) ∈ V
4		fvex 6424	. . 3 ⊢ (2nd ‘𝐴) ∈ V
5	3, 4	opth 5135	. 2 ⊢ (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ = ⟨𝐵, 𝐶⟩ ↔ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶))
6	2, 5	syl6bb 279	1 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → (𝐴 = ⟨𝐵, 𝐶⟩ ↔ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶)))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 385   = wceq 1653   ∈ wcel 2157  ⟨cop 4374   × cxp 5310  ‘cfv 6101  1^st c1st 7399  2^nd c2nd 7400

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-iota 6064  df-fun 6103  df-fv 6109  df-1st 7401  df-2nd 7402

This theorem is referenced by:  eqop2  7444  op1steq  7445  el2xptp0  7447  lsmhash  18431  txhmeo  21935  ptuncnv  21939  wlkcomp  26880  clwlkcomp  27033  f1od2  30017  esum2dlem  30670  poimirlem22  33920  rngosn3  34210  dvhb1dimN  37007