Theorem brwdom 8355

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 3185	. 2 ⊢ (𝑌 ∈ 𝑉 → 𝑌 ∈ V)
2		relwdom 8354	. . . . 5 ⊢ Rel ≼*
3	2	brrelexi 5082	. . . 4 ⊢ (𝑋 ≼* 𝑌 → 𝑋 ∈ V)
4	3	a1i 11	. . 3 ⊢ (𝑌 ∈ V → (𝑋 ≼* 𝑌 → 𝑋 ∈ V))
5		0ex 4718	. . . . . 6 ⊢ ∅ ∈ V
6		eleq1a 2683	. . . . . 6 ⊢ (∅ ∈ V → (𝑋 = ∅ → 𝑋 ∈ V))
7	5, 6	ax-mp 5	. . . . 5 ⊢ (𝑋 = ∅ → 𝑋 ∈ V)
8		forn 6031	. . . . . . 7 ⊢ (𝑧:𝑌–onto→𝑋 → ran 𝑧 = 𝑋)
9		vex 3176	. . . . . . . 8 ⊢ 𝑧 ∈ V
10	9	rnex 6992	. . . . . . 7 ⊢ ran 𝑧 ∈ V
11	8, 10	syl6eqelr 2697	. . . . . 6 ⊢ (𝑧:𝑌–onto→𝑋 → 𝑋 ∈ V)
12	11	exlimiv 1845	. . . . 5 ⊢ (∃𝑧 𝑧:𝑌–onto→𝑋 → 𝑋 ∈ V)
13	7, 12	jaoi 393	. . . 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 2614	. . . . . 6 ⊢ (𝑥 = 𝑋 → (𝑥 = ∅ ↔ 𝑋 = ∅))
16		foeq3 6026	. . . . . . 7 ⊢ (𝑥 = 𝑋 → (𝑧:𝑦–onto→𝑥 ↔ 𝑧:𝑦–onto→𝑋))
17	16	exbidv 1837	. . . . . 6 ⊢ (𝑥 = 𝑋 → (∃𝑧 𝑧:𝑦–onto→𝑥 ↔ ∃𝑧 𝑧:𝑦–onto→𝑋))
18	15, 17	orbi12d 742	. . . . 5 ⊢ (𝑥 = 𝑋 → ((𝑥 = ∅ ∨ ∃𝑧 𝑧:𝑦–onto→𝑥) ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑦–onto→𝑋)))
19		foeq2 6025	. . . . . . 7 ⊢ (𝑦 = 𝑌 → (𝑧:𝑦–onto→𝑋 ↔ 𝑧:𝑌–onto→𝑋))
20	19	exbidv 1837	. . . . . 6 ⊢ (𝑦 = 𝑌 → (∃𝑧 𝑧:𝑦–onto→𝑋 ↔ ∃𝑧 𝑧:𝑌–onto→𝑋))
21	20	orbi2d 734	. . . . 5 ⊢ (𝑦 = 𝑌 → ((𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑦–onto→𝑋) ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋)))
22		df-wdom 8347	. . . . 5 ⊢ ≼* = {⟨𝑥, 𝑦⟩ ∣ (𝑥 = ∅ ∨ ∃𝑧 𝑧:𝑦–onto→𝑥)}
23	18, 21, 22	brabg 4919	. . . 4 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋 ≼* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋)))
24	23	expcom 450	. . 3 ⊢ (𝑌 ∈ V → (𝑋 ∈ V → (𝑋 ≼* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌–onto→𝑋))))
25	4, 14, 24	pm5.21ndd 368	. 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 195   ∨ wo 382   = wceq 1475  ∃wex 1695   ∈ wcel 1977  Vcvv 3173  ∅c0 3874   class class class wbr 4583  ran crn 5039  –onto→wfo 5802   ≼^* cwdom 8345

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-dm 5048  df-rn 5049  df-fn 5807  df-fo 5810  df-wdom 8347

This theorem is referenced by:  brwdomi  8356  brwdomn0  8357  0wdom  8358  fowdom  8359  domwdom  8362  wdomnumr  8770