Theorem dmprop 6183

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 6181	. 2 ⊢ ((𝐵 ∈ V ∧ 𝐷 ∈ V) → dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {𝐴, 𝐶})
4	1, 2, 3	mp2an 693	1 ⊢ dom {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = {𝐴, 𝐶}

Syntax hints:   = wceq 1542   ∈ wcel 2114  Vcvv 3442  {cpr 4584  ⟨cop 4588  dom cdm 5632

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-dm 5642

This theorem is referenced by:  dmtpop  6184  funtp  6557  fpr  7109  fnprb  7164  hashfun  14372  umgr2v2evd2  29613  ex-dm  30526