Theorem op1sta 6189

Description: Extract the first member of an ordered pair. (See op2nda 6192 to extract the second member, op1stb 5424 for an alternate version, and op1st 7950 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 6180	. . 3 ⊢ dom {⟨𝐴, 𝐵⟩} = {𝐴}
3	2	unieqi 4862	. 2 ⊢ ∪ dom {⟨𝐴, 𝐵⟩} = ∪ {𝐴}
4		cnvsn.1	. . 3 ⊢ 𝐴 ∈ V
5	4	unisn 4869	. 2 ⊢ ∪ {𝐴} = 𝐴
6	3, 5	eqtri 2759	1 ⊢ ∪ dom {⟨𝐴, 𝐵⟩} = 𝐴

Syntax hints:   = wceq 1542   ∈ wcel 2114  Vcvv 3429  {csn 4567  ⟨cop 4573  ∪ cuni 4850  dom cdm 5631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-dm 5641

This theorem is referenced by:  elxp4  7873  op1st  7950  fo1st  7962  f1stres  7966  xpassen  9009  xpdom2  9010