Theorem brwdom 9574

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 3475	. 2 ⊢ (𝑌 ∈ 𝑉 → 𝑌 ∈ V)
2		relwdom 9573	. . . . 5 ⊢ Rel ≼*
3	2	brrelex1i 5711	. . . 4 ⊢ (𝑋 ≼* 𝑌 → 𝑋 ∈ V)
4	3	a1i 11	. . 3 ⊢ (𝑌 ∈ V → (𝑋 ≼* 𝑌 → 𝑋 ∈ V))
5		0ex 5264	. . . . . 6 ⊢ ∅ ∈ V
6		eleq1a 2859	. . . . . 6 ⊢ (∅ ∈ V → (𝑋 = ∅ → 𝑋 ∈ V))
7	5, 6	ax-mp 5	. . . . 5 ⊢ (𝑋 = ∅ → 𝑋 ∈ V)
8		forn 6810	. . . . . . 7 ⊢ (𝑧:𝑌–onto→𝑋 → ran 𝑧 = 𝑋)
9		vex 3458	. . . . . . . 8 ⊢ 𝑧 ∈ V
10	9	rnex 7932	. . . . . . 7 ⊢ ran 𝑧 ∈ V
11	8, 10	eqeltrrdi 2873	. . . . . 6 ⊢ (𝑧:𝑌–onto→𝑋 → 𝑋 ∈ V)
12	11	exlimiv 1964	. . . . 5 ⊢ (∃𝑧 𝑧:𝑌–onto→𝑋 → 𝑋 ∈ V)
13	7, 12	jaoi 871	. . . 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 2767	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 = ∅ ↔ 𝑋 = ∅))
16		foeq3 6805	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑧:𝑦–onto→𝑥 ↔ 𝑧:𝑦–onto→𝑋))
17	16	exbidv 1955	. . . . . 6 ⊢ (𝑥 = 𝑋 → (∃𝑧 𝑧:𝑦–onto→𝑥 ↔ ∃𝑧 𝑧:𝑦–onto→𝑋))
18	15, 17	orbi12d 932	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝑥 = ∅ ∨ ∃𝑧 𝑧:𝑦–onto→𝑥) ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑦–onto→𝑋)))
19		foeq2 6804	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑧:𝑦–onto→𝑋 ↔ 𝑧:𝑌–onto→𝑋))
20	19	exbidv 1955	. . . . . 6 ⊢ (𝑦 = 𝑌 → (∃𝑧 𝑧:𝑦–onto→𝑋 ↔ ∃𝑧 𝑧:𝑌–onto→𝑋))
21	20	orbi2d 929	. . . . 5 ⊢ (𝑦 = 𝑌 → ((𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑦–onto→𝑋) ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋)))
22		df-wdom 9572	. . . . 5 ⊢ ≼* = {⟨𝑥, 𝑦⟩ ∣ (𝑥 = ∅ ∨ ∃𝑧 𝑧:𝑦–onto→𝑥)}
23	18, 21, 22	brabg 5518	. . . 4 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋 ≼* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋)))
24	23	expcom 419	. . 3 ⊢ (𝑌 ∈ V → (𝑋 ∈ V → (𝑋 ≼* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋))))
25	4, 14, 24	pm5.21ndd 382	. 2 ⊢ (𝑌 ∈ V → (𝑋 ≼* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋)))
26	1, 25	syl 18	1 ⊢ (𝑌 ∈ 𝑉 → (𝑋 ≼* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∨ wo 861   = wceq 1569  ∃wex 1811   ∈ wcel 2146  Vcvv 3454  ∅c0 4281   class class class wbr 5106  ran crn 5656  –onto→wfo 6545   ≼^* cwdom 9571

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395  ax-un 7761

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-xp 5661  df-rel 5662  df-cnv 5663  df-dm 5665  df-rn 5666  df-fn 6550  df-fo 6553  df-wdom 9572

This theorem is used by:  brwdomi  9575  brwdomn0  9576  0wdom  9577  fowdom  9578  domwdom  9581  wdomnumr  10156