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:𝑌–onto→𝑋))

Distinct variable groups:   𝑧,𝑋   𝑧,𝑌

Proof of Theorem brwdomi

Step	Hyp	Ref	Expression
1		relwdom 9597	. . . 4 ⊢ Rel ≼*
2	1	brrelex2i 5739	. . 3 ⊢ (𝑋 ≼* 𝑌 → 𝑌 ∈ V)
3		brwdom 9598	. . 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 845   = wceq 1533  ∃wex 1773   ∈ wcel 2098  Vcvv 3473  ∅c0 4326   class class class wbr 5152  –onto→wfo 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