Theorem op1sta 5210

Description: Extract the first member of an ordered pair. (See op2nda 5213 to extract the second member and op1stb 4569 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 5202	. . 3 ⊢ dom {⟨𝐴, 𝐵⟩} = {𝐴}
3	2	unieqi 3898	. 2 ⊢ ∪ dom {⟨𝐴, 𝐵⟩} = ∪ {𝐴}
4		cnvsn.1	. . 3 ⊢ 𝐴 ∈ V
5	4	unisn 3904	. 2 ⊢ ∪ {𝐴} = 𝐴
6	3, 5	eqtri 2250	1 ⊢ ∪ dom {⟨𝐴, 𝐵⟩} = 𝐴

Syntax hints:   = wceq 1395   ∈ wcel 2200  Vcvv 2799  {csn 3666  ⟨cop 3669  ∪ cuni 3888  dom cdm 4719

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-dm 4729

This theorem is referenced by:  op1st  6292  fo1st  6303  f1stres  6305  xpassen  6989  xpdom2  6990