Theorem eqop 6075

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 6073	. . 3 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
2	1	eqeq1d 2148	. 2 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → (𝐴 = ⟨𝐵, 𝐶⟩ ↔ ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ = ⟨𝐵, 𝐶⟩))
3		1stexg 6065	. . 3 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → (1st ‘𝐴) ∈ V)
4		2ndexg 6066	. . 3 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → (2nd ‘𝐴) ∈ V)
5		opthg 4160	. . 3 ⊢ (((1st ‘𝐴) ∈ V ∧ (2nd ‘𝐴) ∈ V) → (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ = ⟨𝐵, 𝐶⟩ ↔ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶)))
6	3, 4, 5	syl2anc 408	. 2 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ = ⟨𝐵, 𝐶⟩ ↔ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶)))
7	2, 6	bitrd 187	1 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → (𝐴 = ⟨𝐵, 𝐶⟩ ↔ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶)))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1331   ∈ wcel 1480  Vcvv 2686  ⟨cop 3530   × cxp 4537  ‘cfv 5123  1^st c1st 6036  2^nd c2nd 6037

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-fo 5129  df-fv 5131  df-1st 6038  df-2nd 6039

This theorem is referenced by:  eqop2  6076  op1steq  6077  f1od2  6132  txhmeo  12488