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Theorem brdom 7952
 Description: Dominance relation. (Contributed by NM, 15-Jun-1998.)
Hypothesis
Ref Expression
bren.1 𝐵 ∈ V
Assertion
Ref Expression
brdom (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1𝐵)
Distinct variable groups:   𝐴,𝑓   𝐵,𝑓

Proof of Theorem brdom
StepHypRef Expression
1 bren.1 . 2 𝐵 ∈ V
2 brdomg 7950 . 2 (𝐵 ∈ V → (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1𝐵))
31, 2ax-mp 5 1 (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1𝐵)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 196  ∃wex 1702   ∈ wcel 1988  Vcvv 3195   class class class wbr 4644  –1-1→wf1 5873   ≼ cdom 7938 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897  ax-un 6934 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-xp 5110  df-rel 5111  df-cnv 5112  df-dm 5114  df-rn 5115  df-fn 5879  df-f 5880  df-f1 5881  df-dom 7942 This theorem is referenced by:  domen  7953  domtr  7994  sbthlem10  8064  1sdom  8148  ac10ct  8842  domtriomlem  9249  2ndcdisj  21240  birthdaylem3  24661
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