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Theorem brdomiOLD 8999
Description: Obsolete version of brdomi 8998 as of 29-Nov-2024. (Contributed by Mario Carneiro, 26-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
brdomiOLD (𝐴𝐵 → ∃𝑓 𝑓:𝐴1-1𝐵)
Distinct variable groups:   𝐴,𝑓   𝐵,𝑓

Proof of Theorem brdomiOLD
StepHypRef Expression
1 reldom 8990 . . . 4 Rel ≼
21brrelex2i 5746 . . 3 (𝐴𝐵𝐵 ∈ V)
3 brdomg 8996 . . 3 (𝐵 ∈ V → (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1𝐵))
42, 3syl 17 . 2 (𝐴𝐵 → (𝐴𝐵 ↔ ∃𝑓 𝑓:𝐴1-1𝐵))
54ibi 267 1 (𝐴𝐵 → ∃𝑓 𝑓:𝐴1-1𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wex 1776  wcel 2106  Vcvv 3478   class class class wbr 5148  1-1wf1 6560  cdom 8982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700  df-fn 6566  df-f 6567  df-f1 6568  df-dom 8986
This theorem is referenced by: (None)
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