Theorem brwdom 9605

Description: Property of weak dominance (definitional form). (Contributed by Stefan O'Rear, 11-Feb-2015.)

Assertion

Ref	Expression
brwdom	⊢ (𝑌 ∈ 𝑉 → (𝑋 ≼* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋)))

Distinct variable groups:   𝑧,𝑋   𝑧,𝑌

Allowed substitution hint:   𝑉(𝑧)

Proof of Theorem brwdom

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3499	. 2 ⊢ (𝑌 ∈ 𝑉 → 𝑌 ∈ V)
2		relwdom 9604	. . . . 5 ⊢ Rel ≼*
3	2	brrelex1i 5745	. . . 4 ⊢ (𝑋 ≼* 𝑌 → 𝑋 ∈ V)
4	3	a1i 11	. . 3 ⊢ (𝑌 ∈ V → (𝑋 ≼* 𝑌 → 𝑋 ∈ V))
5		0ex 5313	. . . . . 6 ⊢ ∅ ∈ V
6		eleq1a 2834	. . . . . 6 ⊢ (∅ ∈ V → (𝑋 = ∅ → 𝑋 ∈ V))
7	5, 6	ax-mp 5	. . . . 5 ⊢ (𝑋 = ∅ → 𝑋 ∈ V)
8		forn 6824	. . . . . . 7 ⊢ (𝑧:𝑌–onto→𝑋 → ran 𝑧 = 𝑋)
9		vex 3482	. . . . . . . 8 ⊢ 𝑧 ∈ V
10	9	rnex 7933	. . . . . . 7 ⊢ ran 𝑧 ∈ V
11	8, 10	eqeltrrdi 2848	. . . . . 6 ⊢ (𝑧:𝑌–onto→𝑋 → 𝑋 ∈ V)
12	11	exlimiv 1928	. . . . 5 ⊢ (∃𝑧 𝑧:𝑌–onto→𝑋 → 𝑋 ∈ V)
13	7, 12	jaoi 857	. . . 4 ⊢ ((𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋) → 𝑋 ∈ V)
14	13	a1i 11	. . 3 ⊢ (𝑌 ∈ V → ((𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋) → 𝑋 ∈ V))
15		eqeq1 2739	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 = ∅ ↔ 𝑋 = ∅))
16		foeq3 6819	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑧:𝑦–onto→𝑥 ↔ 𝑧:𝑦–onto→𝑋))
17	16	exbidv 1919	. . . . . 6 ⊢ (𝑥 = 𝑋 → (∃𝑧 𝑧:𝑦–onto→𝑥 ↔ ∃𝑧 𝑧:𝑦–onto→𝑋))
18	15, 17	orbi12d 918	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝑥 = ∅ ∨ ∃𝑧 𝑧:𝑦–onto→𝑥) ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑦–onto→𝑋)))
19		foeq2 6818	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑧:𝑦–onto→𝑋 ↔ 𝑧:𝑌–onto→𝑋))
20	19	exbidv 1919	. . . . . 6 ⊢ (𝑦 = 𝑌 → (∃𝑧 𝑧:𝑦–onto→𝑋 ↔ ∃𝑧 𝑧:𝑌–onto→𝑋))
21	20	orbi2d 915	. . . . 5 ⊢ (𝑦 = 𝑌 → ((𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑦–onto→𝑋) ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋)))
22		df-wdom 9603	. . . . 5 ⊢ ≼* = {⟨𝑥, 𝑦⟩ ∣ (𝑥 = ∅ ∨ ∃𝑧 𝑧:𝑦–onto→𝑥)}
23	18, 21, 22	brabg 5549	. . . 4 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋 ≼* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋)))
24	23	expcom 413	. . 3 ⊢ (𝑌 ∈ V → (𝑋 ∈ V → (𝑋 ≼* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋))))
25	4, 14, 24	pm5.21ndd 379	. 2 ⊢ (𝑌 ∈ V → (𝑋 ≼* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋)))
26	1, 25	syl 17	1 ⊢ (𝑌 ∈ 𝑉 → (𝑋 ≼* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋)))

Syntax hints:   → wi 4   ↔ wb 206   ∨ wo 847   = wceq 1537  ∃wex 1776   ∈ wcel 2106  Vcvv 3478  ∅c0 4339   class class class wbr 5148  ran crn 5690  –onto→wfo 6561   ≼^* cwdom 9602

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700  df-fn 6566  df-fo 6569  df-wdom 9603

This theorem is referenced by:  brwdomi  9606  brwdomn0  9607  0wdom  9608  fowdom  9609  domwdom  9612  wdomnumr  10102