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Theorem brwdom 8637
Description: Property of weak dominance (definitional form). (Contributed by Stefan O'Rear, 11-Feb-2015.)
Assertion
Ref Expression
brwdom (𝑌𝑉 → (𝑋* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋)))
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.
StepHypRef Expression
1 elex 3352 . 2 (𝑌𝑉𝑌 ∈ V)
2 relwdom 8636 . . . . 5 Rel ≼*
32brrelexi 5315 . . . 4 (𝑋* 𝑌𝑋 ∈ V)
43a1i 11 . . 3 (𝑌 ∈ V → (𝑋* 𝑌𝑋 ∈ V))
5 0ex 4942 . . . . . 6 ∅ ∈ V
6 eleq1a 2834 . . . . . 6 (∅ ∈ V → (𝑋 = ∅ → 𝑋 ∈ V))
75, 6ax-mp 5 . . . . 5 (𝑋 = ∅ → 𝑋 ∈ V)
8 forn 6279 . . . . . . 7 (𝑧:𝑌onto𝑋 → ran 𝑧 = 𝑋)
9 vex 3343 . . . . . . . 8 𝑧 ∈ V
109rnex 7265 . . . . . . 7 ran 𝑧 ∈ V
118, 10syl6eqelr 2848 . . . . . 6 (𝑧:𝑌onto𝑋𝑋 ∈ V)
1211exlimiv 2007 . . . . 5 (∃𝑧 𝑧:𝑌onto𝑋𝑋 ∈ V)
137, 12jaoi 393 . . . 4 ((𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋) → 𝑋 ∈ V)
1413a1i 11 . . 3 (𝑌 ∈ V → ((𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋) → 𝑋 ∈ V))
15 eqeq1 2764 . . . . . 6 (𝑥 = 𝑋 → (𝑥 = ∅ ↔ 𝑋 = ∅))
16 foeq3 6274 . . . . . . 7 (𝑥 = 𝑋 → (𝑧:𝑦onto𝑥𝑧:𝑦onto𝑋))
1716exbidv 1999 . . . . . 6 (𝑥 = 𝑋 → (∃𝑧 𝑧:𝑦onto𝑥 ↔ ∃𝑧 𝑧:𝑦onto𝑋))
1815, 17orbi12d 748 . . . . 5 (𝑥 = 𝑋 → ((𝑥 = ∅ ∨ ∃𝑧 𝑧:𝑦onto𝑥) ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑦onto𝑋)))
19 foeq2 6273 . . . . . . 7 (𝑦 = 𝑌 → (𝑧:𝑦onto𝑋𝑧:𝑌onto𝑋))
2019exbidv 1999 . . . . . 6 (𝑦 = 𝑌 → (∃𝑧 𝑧:𝑦onto𝑋 ↔ ∃𝑧 𝑧:𝑌onto𝑋))
2120orbi2d 740 . . . . 5 (𝑦 = 𝑌 → ((𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑦onto𝑋) ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋)))
22 df-wdom 8629 . . . . 5 * = {⟨𝑥, 𝑦⟩ ∣ (𝑥 = ∅ ∨ ∃𝑧 𝑧:𝑦onto𝑥)}
2318, 21, 22brabg 5144 . . . 4 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋)))
2423expcom 450 . . 3 (𝑌 ∈ V → (𝑋 ∈ V → (𝑋* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋))))
254, 14, 24pm5.21ndd 368 . 2 (𝑌 ∈ V → (𝑋* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋)))
261, 25syl 17 1 (𝑌𝑉 → (𝑋* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382   = wceq 1632  wex 1853  wcel 2139  Vcvv 3340  c0 4058   class class class wbr 4804  ran crn 5267  ontowfo 6047  * cwdom 8627
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-xp 5272  df-rel 5273  df-cnv 5274  df-dm 5276  df-rn 5277  df-fn 6052  df-fo 6055  df-wdom 8629
This theorem is referenced by:  brwdomi  8638  brwdomn0  8639  0wdom  8640  fowdom  8641  domwdom  8644  wdomnumr  9077
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