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Theorem brwdomi 9599
Description: Property of weak dominance, forward direction only. (Contributed by Mario Carneiro, 5-May-2015.)
Assertion
Ref Expression
brwdomi (𝑋* 𝑌 → (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋))
Distinct variable groups:   𝑧,𝑋   𝑧,𝑌

Proof of Theorem brwdomi
StepHypRef Expression
1 relwdom 9597 . . . 4 Rel ≼*
21brrelex2i 5739 . . 3 (𝑋* 𝑌𝑌 ∈ V)
3 brwdom 9598 . . 3 (𝑌 ∈ V → (𝑋* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋)))
42, 3syl 17 . 2 (𝑋* 𝑌 → (𝑋* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋)))
54ibi 266 1 (𝑋* 𝑌 → (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wo 845   = wceq 1533  wex 1773  wcel 2098  Vcvv 3473  c0 4326   class class class wbr 5152  ontowfo 6551  * cwdom 9595
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-xp 5688  df-rel 5689  df-cnv 5690  df-dm 5692  df-rn 5693  df-fn 6556  df-fo 6559  df-wdom 9596
This theorem is referenced by:  numwdom  10090
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