Theorem op1sta 5085

Description: Extract the first member of an ordered pair. (See op2nda 5088 to extract the second member and op1stb 4456 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 5077	. . 3 ⊢ dom {⟨𝐴, 𝐵⟩} = {𝐴}
3	2	unieqi 3799	. 2 ⊢ ∪ dom {⟨𝐴, 𝐵⟩} = ∪ {𝐴}
4		cnvsn.1	. . 3 ⊢ 𝐴 ∈ V
5	4	unisn 3805	. 2 ⊢ ∪ {𝐴} = 𝐴
6	3, 5	eqtri 2186	1 ⊢ ∪ dom {⟨𝐴, 𝐵⟩} = 𝐴

Syntax hints:   = wceq 1343   ∈ wcel 2136  Vcvv 2726  {csn 3576  ⟨cop 3579  ∪ cuni 3789  dom cdm 4604

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-dm 4614

This theorem is referenced by:  op1st  6114  fo1st  6125  f1stres  6127  xpassen  6796  xpdom2  6797