Theorem dmprop 5209

Description: The domain of an unordered pair of ordered pairs. (Contributed by NM, 13-Sep-2011.)

Hypotheses

Ref	Expression
dmsnop.1	⊢ 𝐵 ∈ V
dmprop.1	⊢ 𝐷 ∈ V

Assertion

Ref	Expression
dmprop	⊢ dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {𝐴, 𝐶}

Proof of Theorem dmprop

Step	Hyp	Ref	Expression
1		dmsnop.1	. 2 ⊢ 𝐵 ∈ V
2		dmprop.1	. 2 ⊢ 𝐷 ∈ V
3		dmpropg 5207	. 2 ⊢ ((𝐵 ∈ V ∧ 𝐷 ∈ V) → dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {𝐴, 𝐶})
4	1, 2, 3	mp2an 426	1 ⊢ dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {𝐴, 𝐶}

Syntax hints:   = wceq 1395   ∈ wcel 2200  Vcvv 2800  {cpr 3668  ⟨cop 3670  dom cdm 4723

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 4205  ax-pow 4262  ax-pr 4297

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-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-br 4087  df-dm 4733

This theorem is referenced by:  dmtpop  5210  funtp  5380  fpr  5831