Theorem brwdomi 9590

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 9588	. . . 4 ⊢ Rel ≼*
2	1	brrelex2i 5722	. . 3 ⊢ (𝑋 ≼* 𝑌 → 𝑌 ∈ V)
3		brwdom 9589	. . 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 267	1 ⊢ (𝑋 ≼* 𝑌 → (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋))

Syntax hints:   → wi 4   ↔ wb 206   ∨ wo 847   = wceq 1539  ∃wex 1778   ∈ wcel 2107  Vcvv 3463  ∅c0 4313   class class class wbr 5123  –onto→wfo 6539   ≼^* cwdom 9586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412  ax-un 7737

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-br 5124  df-opab 5186  df-xp 5671  df-rel 5672  df-cnv 5673  df-dm 5675  df-rn 5676  df-fn 6544  df-fo 6547  df-wdom 9587

This theorem is referenced by:  numwdom  10081