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Theorem brdomi 7926
Description: Dominance relation. (Contributed by Mario Carneiro, 26-Apr-2015.)
Assertion
Ref Expression
brdomi (𝐴𝐵 → ∃𝑓 𝑓:𝐴1-1𝐵)
Distinct variable groups:   𝐴,𝑓   𝐵,𝑓

Proof of Theorem brdomi
StepHypRef Expression
1 reldom 7921 . . . 4 Rel ≼
21brrelex2i 5129 . . 3 (𝐴𝐵𝐵 ∈ V)
3 brdomg 7925 . . 3 (𝐵 ∈ V → (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1𝐵))
42, 3syl 17 . 2 (𝐴𝐵 → (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1𝐵))
54ibi 256 1 (𝐴𝐵 → ∃𝑓 𝑓:𝐴1-1𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wex 1701  wcel 1987  Vcvv 3190   class class class wbr 4623  1-1wf1 5854  cdom 7913
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-xp 5090  df-rel 5091  df-cnv 5092  df-dm 5094  df-rn 5095  df-fn 5860  df-f 5861  df-f1 5862  df-dom 7917
This theorem is referenced by:  ctex  7930  2dom  7989  xpdom2  8015  domunsncan  8020  fodomr  8071  domssex  8081  sucdom2  8116  hartogslem1  8407  infdifsn  8514  acndom  8834  acndom2  8837  fictb  9027  fin23lem41  9134  iundom2g  9322  pwfseq  9446  omssubadd  30185
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