Theorem eqop 6178

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 6176	. . 3 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → 𝐴 = ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩)
2	1	eqeq1d 2186	. 2 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → (𝐴 = ⟨𝐵, 𝐶⟩ ↔ ⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ = ⟨𝐵, 𝐶⟩))
3		1stexg 6168	. . 3 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → (1st ‘𝐴) ∈ V)
4		2ndexg 6169	. . 3 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → (2nd ‘𝐴) ∈ V)
5		opthg 4239	. . 3 ⊢ (((1st ‘𝐴) ∈ V ∧ (2nd ‘𝐴) ∈ V) → (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ = ⟨𝐵, 𝐶⟩ ↔ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶)))
6	3, 4, 5	syl2anc 411	. 2 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → (⟨(1st ‘𝐴), (2nd ‘𝐴)⟩ = ⟨𝐵, 𝐶⟩ ↔ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶)))
7	2, 6	bitrd 188	1 ⊢ (𝐴 ∈ (𝑉 × 𝑊) → (𝐴 = ⟨𝐵, 𝐶⟩ ↔ ((1st ‘𝐴) = 𝐵 ∧ (2nd ‘𝐴) = 𝐶)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1353   ∈ wcel 2148  Vcvv 2738  ⟨cop 3596   × cxp 4625  ‘cfv 5217  1^st c1st 6139  2^nd c2nd 6140

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-sbc 2964  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-opab 4066  df-mpt 4067  df-id 4294  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-fo 5223  df-fv 5225  df-1st 6141  df-2nd 6142

This theorem is referenced by:  eqop2  6179  op1steq  6180  f1od2  6236  txhmeo  13822