Theorem op1sta 5148

Description: Extract the first member of an ordered pair. (See op2nda 5151 to extract the second member and op1stb 4510 for an alternate 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 5140	. . 3 ⊢ dom {⟨𝐴, 𝐵⟩} = {𝐴}
3	2	unieqi 3846	. 2 ⊢ ∪ dom {⟨𝐴, 𝐵⟩} = ∪ {𝐴}
4		cnvsn.1	. . 3 ⊢ 𝐴 ∈ V
5	4	unisn 3852	. 2 ⊢ ∪ {𝐴} = 𝐴
6	3, 5	eqtri 2214	1 ⊢ ∪ dom {⟨𝐴, 𝐵⟩} = 𝐴

Syntax hints:   = wceq 1364   ∈ wcel 2164  Vcvv 2760  {csn 3619  ⟨cop 3622  ∪ cuni 3836  dom cdm 4660

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-dm 4670

This theorem is referenced by:  op1st  6201  fo1st  6212  f1stres  6214  xpassen  6886  xpdom2  6887