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Theorem brwdomi 8829
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 8827 . . . 4 Rel ≼*
21brrelex2i 5460 . . 3 (𝑋* 𝑌𝑌 ∈ V)
3 brwdom 8828 . . 3 (𝑌 ∈ V → (𝑋* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋)))
42, 3syl 17 . 2 (𝑋* 𝑌 → (𝑋* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋)))
54ibi 259 1 (𝑋* 𝑌 → (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wo 833   = wceq 1507  wex 1742  wcel 2050  Vcvv 3415  c0 4180   class class class wbr 4930  ontowfo 6188  * cwdom 8818
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5061  ax-nul 5068  ax-pr 5187  ax-un 7281
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2583  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-nul 4181  df-if 4352  df-sn 4443  df-pr 4445  df-op 4449  df-uni 4714  df-br 4931  df-opab 4993  df-xp 5414  df-rel 5415  df-cnv 5416  df-dm 5418  df-rn 5419  df-fn 6193  df-fo 6196  df-wdom 8820
This theorem is referenced by:  numwdom  9281
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