Theorem op1sta 6231

Description: Extract the first member of an ordered pair. (See op2nda 6234 to extract the second member, op1stb 5473 for an alternate version, and op1st 8002 for the preferred version.) (Contributed by Raph Levien, 4-Dec-2003.)

Hypotheses

Ref	Expression
cnvsn.1	⊢ 𝐴 ∈ V
cnvsn.2	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
op1sta	⊢ ∪ dom {⟨𝐴, 𝐵⟩} = 𝐴

Proof of Theorem op1sta

Step	Hyp	Ref	Expression
1		cnvsn.2	. . . 4 ⊢ 𝐵 ∈ V
2	1	dmsnop 6222	. . 3 ⊢ dom {⟨𝐴, 𝐵⟩} = {𝐴}
3	2	unieqi 4921	. 2 ⊢ ∪ dom {⟨𝐴, 𝐵⟩} = ∪ {𝐴}
4		cnvsn.1	. . 3 ⊢ 𝐴 ∈ V
5	4	unisn 4930	. 2 ⊢ ∪ {𝐴} = 𝐴
6	3, 5	eqtri 2753	1 ⊢ ∪ dom {⟨𝐴, 𝐵⟩} = 𝐴

Syntax hints:   = wceq 1533   ∈ wcel 2098  Vcvv 3461  {csn 4630  ⟨cop 4636  ∪ cuni 4909  dom cdm 5678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-dm 5688

This theorem is referenced by:  elxp4  7930  op1st  8002  fo1st  8014  f1stres  8018  xpassen  9091  xpdom2  9092