Theorem brwdomi 9257

Description: Property of weak dominance, forward direction only. (Contributed by Mario Carneiro, 5-May-2015.)

Assertion

Ref	Expression
brwdomi	⊢ (𝑋 ≼* 𝑌 → (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋))

Distinct variable groups:   𝑧,𝑋   𝑧,𝑌

Proof of Theorem brwdomi

Step	Hyp	Ref	Expression
1		relwdom 9255	. . . 4 ⊢ Rel ≼*
2	1	brrelex2i 5635	. . 3 ⊢ (𝑋 ≼* 𝑌 → 𝑌 ∈ V)
3		brwdom 9256	. . 3 ⊢ (𝑌 ∈ V → (𝑋 ≼* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋)))
4	2, 3	syl 17	. 2 ⊢ (𝑋 ≼* 𝑌 → (𝑋 ≼* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋)))
5	4	ibi 266	1 ⊢ (𝑋 ≼* 𝑌 → (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋))

Syntax hints:   → wi 4   ↔ wb 205   ∨ wo 843   = wceq 1539  ∃wex 1783   ∈ wcel 2108  Vcvv 3422  ∅c0 4253   class class class wbr 5070  –onto→wfo 6416   ≼^* cwdom 9253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-xp 5586  df-rel 5587  df-cnv 5588  df-dm 5590  df-rn 5591  df-fn 6421  df-fo 6424  df-wdom 9254

This theorem is referenced by:  numwdom  9746