Theorem brwdomi 9454

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 9452	. . . 4 ⊢ Rel ≼*
2	1	brrelex2i 5671	. . 3 ⊢ (𝑋 ≼* 𝑌 → 𝑌 ∈ V)
3		brwdom 9453	. . 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 1541  ∃wex 1780   ∈ wcel 2111  Vcvv 3436  ∅c0 4280   class class class wbr 5089  –onto→wfo 6479   ≼^* cwdom 9450

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-xp 5620  df-rel 5621  df-cnv 5622  df-dm 5624  df-rn 5625  df-fn 6484  df-fo 6487  df-wdom 9451

This theorem is referenced by:  numwdom  9950