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Theorem brdom 8172
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 8170 . 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 197  wex 1874  wcel 2155  Vcvv 3350   class class class wbr 4809  1-1wf1 6065  cdom 8158
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-xp 5283  df-rel 5284  df-cnv 5285  df-dm 5287  df-rn 5288  df-fn 6071  df-f 6072  df-f1 6073  df-dom 8162
This theorem is referenced by:  domen  8173  domtr  8213  sbthlem10  8286  1sdom  8370  ac10ct  9108  domtriomlem  9517  2ndcdisj  21539  birthdaylem3  24971
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