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Theorem brwdom 8828
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 3433 . 2 (𝑌𝑉𝑌 ∈ V)
2 relwdom 8827 . . . . 5 Rel ≼*
32brrelex1i 5459 . . . 4 (𝑋* 𝑌𝑋 ∈ V)
43a1i 11 . . 3 (𝑌 ∈ V → (𝑋* 𝑌𝑋 ∈ V))
5 0ex 5069 . . . . . 6 ∅ ∈ V
6 eleq1a 2861 . . . . . 6 (∅ ∈ V → (𝑋 = ∅ → 𝑋 ∈ V))
75, 6ax-mp 5 . . . . 5 (𝑋 = ∅ → 𝑋 ∈ V)
8 forn 6424 . . . . . . 7 (𝑧:𝑌onto𝑋 → ran 𝑧 = 𝑋)
9 vex 3418 . . . . . . . 8 𝑧 ∈ V
109rnex 7434 . . . . . . 7 ran 𝑧 ∈ V
118, 10syl6eqelr 2875 . . . . . 6 (𝑧:𝑌onto𝑋𝑋 ∈ V)
1211exlimiv 1889 . . . . 5 (∃𝑧 𝑧:𝑌onto𝑋𝑋 ∈ V)
137, 12jaoi 843 . . . 4 ((𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋) → 𝑋 ∈ V)
1413a1i 11 . . 3 (𝑌 ∈ V → ((𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋) → 𝑋 ∈ V))
15 eqeq1 2782 . . . . . 6 (𝑥 = 𝑋 → (𝑥 = ∅ ↔ 𝑋 = ∅))
16 foeq3 6419 . . . . . . 7 (𝑥 = 𝑋 → (𝑧:𝑦onto𝑥𝑧:𝑦onto𝑋))
1716exbidv 1880 . . . . . 6 (𝑥 = 𝑋 → (∃𝑧 𝑧:𝑦onto𝑥 ↔ ∃𝑧 𝑧:𝑦onto𝑋))
1815, 17orbi12d 902 . . . . 5 (𝑥 = 𝑋 → ((𝑥 = ∅ ∨ ∃𝑧 𝑧:𝑦onto𝑥) ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑦onto𝑋)))
19 foeq2 6418 . . . . . . 7 (𝑦 = 𝑌 → (𝑧:𝑦onto𝑋𝑧:𝑌onto𝑋))
2019exbidv 1880 . . . . . 6 (𝑦 = 𝑌 → (∃𝑧 𝑧:𝑦onto𝑋 ↔ ∃𝑧 𝑧:𝑌onto𝑋))
2120orbi2d 899 . . . . 5 (𝑦 = 𝑌 → ((𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑦onto𝑋) ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋)))
22 df-wdom 8820 . . . . 5 * = {⟨𝑥, 𝑦⟩ ∣ (𝑥 = ∅ ∨ ∃𝑧 𝑧:𝑦onto𝑥)}
2318, 21, 22brabg 5281 . . . 4 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋)))
2423expcom 406 . . 3 (𝑌 ∈ V → (𝑋 ∈ V → (𝑋* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋))))
254, 14, 24pm5.21ndd 372 . 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 198  wo 833   = wceq 1507  wex 1742  wcel 2050  Vcvv 3415  c0 4180   class class class wbr 4930  ran crn 5409  ontowfo 6188  * cwdom 8818
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5061  ax-nul 5068  ax-pr 5187  ax-un 7281
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2583  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-nul 4181  df-if 4352  df-sn 4443  df-pr 4445  df-op 4449  df-uni 4714  df-br 4931  df-opab 4993  df-xp 5414  df-rel 5415  df-cnv 5416  df-dm 5418  df-rn 5419  df-fn 6193  df-fo 6196  df-wdom 8820
This theorem is referenced by:  brwdomi  8829  brwdomn0  8830  0wdom  8831  fowdom  8832  domwdom  8835  wdomnumr  9286
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