Theorem op1sta 6245

Description: Extract the first member of an ordered pair. (See op2nda 6248 to extract the second member, op1stb 5447 for an alternate version, and op1st 8043 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 6236	. . 3 ⊢ dom {⟨𝐴, 𝐵⟩} = {𝐴}
3	2	unieqi 4884	. 2 ⊢ ∪ dom {⟨𝐴, 𝐵⟩} = ∪ {𝐴}
4		cnvsn.1	. . 3 ⊢ 𝐴 ∈ V
5	4	unisn 4891	. 2 ⊢ ∪ {𝐴} = 𝐴
6	3, 5	eqtri 2786	1 ⊢ ∪ dom {⟨𝐴, 𝐵⟩} = 𝐴

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2147  Vcvv 3454  {csn 4588  ⟨cop 4840  ∪ cuni 4872  dom cdm 5658

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-dm 5668

This theorem is used by:  elxp4  7968  op1st  8043  fo1st  8055  f1stres  8059  xpassen  9128  xpdom2  9129