Theorem brwdom 9608

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 3500	. 2 ⊢ (𝑌 ∈ 𝑉 → 𝑌 ∈ V)
2		relwdom 9607	. . . . 5 ⊢ Rel ≼*
3	2	brrelex1i 5740	. . . 4 ⊢ (𝑋 ≼* 𝑌 → 𝑋 ∈ V)
4	3	a1i 11	. . 3 ⊢ (𝑌 ∈ V → (𝑋 ≼* 𝑌 → 𝑋 ∈ V))
5		0ex 5306	. . . . . 6 ⊢ ∅ ∈ V
6		eleq1a 2835	. . . . . 6 ⊢ (∅ ∈ V → (𝑋 = ∅ → 𝑋 ∈ V))
7	5, 6	ax-mp 5	. . . . 5 ⊢ (𝑋 = ∅ → 𝑋 ∈ V)
8		forn 6822	. . . . . . 7 ⊢ (𝑧:𝑌–onto→𝑋 → ran 𝑧 = 𝑋)
9		vex 3483	. . . . . . . 8 ⊢ 𝑧 ∈ V
10	9	rnex 7933	. . . . . . 7 ⊢ ran 𝑧 ∈ V
11	8, 10	eqeltrrdi 2849	. . . . . 6 ⊢ (𝑧:𝑌–onto→𝑋 → 𝑋 ∈ V)
12	11	exlimiv 1929	. . . . 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 2740	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 = ∅ ↔ 𝑋 = ∅))
16		foeq3 6817	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑧:𝑦–onto→𝑥 ↔ 𝑧:𝑦–onto→𝑋))
17	16	exbidv 1920	. . . . . 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 6816	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑧:𝑦–onto→𝑋 ↔ 𝑧:𝑌–onto→𝑋))
20	19	exbidv 1920	. . . . . 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 9606	. . . . 5 ⊢ ≼* = {⟨𝑥, 𝑦⟩ ∣ (𝑥 = ∅ ∨ ∃𝑧 𝑧:𝑦–onto→𝑥)}
23	18, 21, 22	brabg 5543	. . . 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 1539  ∃wex 1778   ∈ wcel 2107  Vcvv 3479  ∅c0 4332   class class class wbr 5142  ran crn 5685  –onto→wfo 6558   ≼^* cwdom 9605

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431  ax-un 7756

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-opab 5205  df-xp 5690  df-rel 5691  df-cnv 5692  df-dm 5694  df-rn 5695  df-fn 6563  df-fo 6566  df-wdom 9606

This theorem is referenced by:  brwdomi  9609  brwdomn0  9610  0wdom  9611  fowdom  9612  domwdom  9615  wdomnumr  10105