Theorem brwdom 9335

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 3451	. 2 ⊢ (𝑌 ∈ 𝑉 → 𝑌 ∈ V)
2		relwdom 9334	. . . . 5 ⊢ Rel ≼*
3	2	brrelex1i 5644	. . . 4 ⊢ (𝑋 ≼* 𝑌 → 𝑋 ∈ V)
4	3	a1i 11	. . 3 ⊢ (𝑌 ∈ V → (𝑋 ≼* 𝑌 → 𝑋 ∈ V))
5		0ex 5232	. . . . . 6 ⊢ ∅ ∈ V
6		eleq1a 2835	. . . . . 6 ⊢ (∅ ∈ V → (𝑋 = ∅ → 𝑋 ∈ V))
7	5, 6	ax-mp 5	. . . . 5 ⊢ (𝑋 = ∅ → 𝑋 ∈ V)
8		forn 6700	. . . . . . 7 ⊢ (𝑧:𝑌–onto→𝑋 → ran 𝑧 = 𝑋)
9		vex 3437	. . . . . . . 8 ⊢ 𝑧 ∈ V
10	9	rnex 7768	. . . . . . 7 ⊢ ran 𝑧 ∈ V
11	8, 10	eqeltrrdi 2849	. . . . . 6 ⊢ (𝑧:𝑌–onto→𝑋 → 𝑋 ∈ V)
12	11	exlimiv 1934	. . . . 5 ⊢ (∃𝑧 𝑧:𝑌–onto→𝑋 → 𝑋 ∈ V)
13	7, 12	jaoi 854	. . . 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 2743	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 = ∅ ↔ 𝑋 = ∅))
16		foeq3 6695	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑧:𝑦–onto→𝑥 ↔ 𝑧:𝑦–onto→𝑋))
17	16	exbidv 1925	. . . . . 6 ⊢ (𝑥 = 𝑋 → (∃𝑧 𝑧:𝑦–onto→𝑥 ↔ ∃𝑧 𝑧:𝑦–onto→𝑋))
18	15, 17	orbi12d 916	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝑥 = ∅ ∨ ∃𝑧 𝑧:𝑦–onto→𝑥) ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑦–onto→𝑋)))
19		foeq2 6694	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑧:𝑦–onto→𝑋 ↔ 𝑧:𝑌–onto→𝑋))
20	19	exbidv 1925	. . . . . 6 ⊢ (𝑦 = 𝑌 → (∃𝑧 𝑧:𝑦–onto→𝑋 ↔ ∃𝑧 𝑧:𝑌–onto→𝑋))
21	20	orbi2d 913	. . . . 5 ⊢ (𝑦 = 𝑌 → ((𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑦–onto→𝑋) ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋)))
22		df-wdom 9333	. . . . 5 ⊢ ≼* = {⟨𝑥, 𝑦⟩ ∣ (𝑥 = ∅ ∨ ∃𝑧 𝑧:𝑦–onto→𝑥)}
23	18, 21, 22	brabg 5453	. . . 4 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋 ≼* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋)))
24	23	expcom 414	. . 3 ⊢ (𝑌 ∈ V → (𝑋 ∈ V → (𝑋 ≼* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋))))
25	4, 14, 24	pm5.21ndd 381	. 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 205   ∨ wo 844   = wceq 1539  ∃wex 1782   ∈ wcel 2107  Vcvv 3433  ∅c0 4257   class class class wbr 5075  ran crn 5591  –onto→wfo 6435   ≼^* cwdom 9332

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2710  ax-sep 5224  ax-nul 5231  ax-pr 5353  ax-un 7597

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2069  df-clab 2717  df-cleq 2731  df-clel 2817  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3435  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-nul 4258  df-if 4461  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4841  df-br 5076  df-opab 5138  df-xp 5596  df-rel 5597  df-cnv 5598  df-dm 5600  df-rn 5601  df-fn 6440  df-fo 6443  df-wdom 9333

This theorem is referenced by:  brwdomi  9336  brwdomn0  9337  0wdom  9338  fowdom  9339  domwdom  9342  wdomnumr  9829