Theorem brwdomi 9327

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 9325	. . . 4 ⊢ Rel ≼*
2	1	brrelex2i 5644	. . 3 ⊢ (𝑋 ≼* 𝑌 → 𝑌 ∈ V)
3		brwdom 9326	. . 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 844   = wceq 1539  ∃wex 1782   ∈ wcel 2106  Vcvv 3432  ∅c0 4256   class class class wbr 5074  –onto→wfo 6431   ≼^* cwdom 9323

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-xp 5595  df-rel 5596  df-cnv 5597  df-dm 5599  df-rn 5600  df-fn 6436  df-fo 6439  df-wdom 9324

This theorem is referenced by:  numwdom  9815