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Theorem brdom 8508
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 8506 . 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 209  wex 1781  wcel 2112  Vcvv 3444   class class class wbr 5033  1-1wf1 6325  cdom 8494
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-xp 5529  df-rel 5530  df-cnv 5531  df-dm 5533  df-rn 5534  df-fn 6331  df-f 6332  df-f1 6333  df-dom 8498
This theorem is referenced by:  domen  8509  domtr  8549  sbthlem10  8624  1sdom  8709  ac10ct  9449  domtriomlem  9857  2ndcdisj  22065  birthdaylem3  25543
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