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Theorem brwdom 9019
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 3462 . 2 (𝑌𝑉𝑌 ∈ V)
2 relwdom 9018 . . . . 5 Rel ≼*
32brrelex1i 5576 . . . 4 (𝑋* 𝑌𝑋 ∈ V)
43a1i 11 . . 3 (𝑌 ∈ V → (𝑋* 𝑌𝑋 ∈ V))
5 0ex 5178 . . . . . 6 ∅ ∈ V
6 eleq1a 2888 . . . . . 6 (∅ ∈ V → (𝑋 = ∅ → 𝑋 ∈ V))
75, 6ax-mp 5 . . . . 5 (𝑋 = ∅ → 𝑋 ∈ V)
8 forn 6572 . . . . . . 7 (𝑧:𝑌onto𝑋 → ran 𝑧 = 𝑋)
9 vex 3447 . . . . . . . 8 𝑧 ∈ V
109rnex 7603 . . . . . . 7 ran 𝑧 ∈ V
118, 10eqeltrrdi 2902 . . . . . 6 (𝑧:𝑌onto𝑋𝑋 ∈ V)
1211exlimiv 1931 . . . . 5 (∃𝑧 𝑧:𝑌onto𝑋𝑋 ∈ V)
137, 12jaoi 854 . . . 4 ((𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋) → 𝑋 ∈ V)
1413a1i 11 . . 3 (𝑌 ∈ V → ((𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋) → 𝑋 ∈ V))
15 eqeq1 2805 . . . . . 6 (𝑥 = 𝑋 → (𝑥 = ∅ ↔ 𝑋 = ∅))
16 foeq3 6567 . . . . . . 7 (𝑥 = 𝑋 → (𝑧:𝑦onto𝑥𝑧:𝑦onto𝑋))
1716exbidv 1922 . . . . . 6 (𝑥 = 𝑋 → (∃𝑧 𝑧:𝑦onto𝑥 ↔ ∃𝑧 𝑧:𝑦onto𝑋))
1815, 17orbi12d 916 . . . . 5 (𝑥 = 𝑋 → ((𝑥 = ∅ ∨ ∃𝑧 𝑧:𝑦onto𝑥) ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑦onto𝑋)))
19 foeq2 6566 . . . . . . 7 (𝑦 = 𝑌 → (𝑧:𝑦onto𝑋𝑧:𝑌onto𝑋))
2019exbidv 1922 . . . . . 6 (𝑦 = 𝑌 → (∃𝑧 𝑧:𝑦onto𝑋 ↔ ∃𝑧 𝑧:𝑌onto𝑋))
2120orbi2d 913 . . . . 5 (𝑦 = 𝑌 → ((𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑦onto𝑋) ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋)))
22 df-wdom 9017 . . . . 5 * = {⟨𝑥, 𝑦⟩ ∣ (𝑥 = ∅ ∨ ∃𝑧 𝑧:𝑦onto𝑥)}
2318, 21, 22brabg 5394 . . . 4 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋)))
2423expcom 417 . . 3 (𝑌 ∈ V → (𝑋 ∈ V → (𝑋* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋))))
254, 14, 24pm5.21ndd 384 . 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 209  wo 844   = wceq 1538  wex 1781  wcel 2112  Vcvv 3444  c0 4246   class class class wbr 5033  ran crn 5524  ontowfo 6326  * cwdom 9016
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-xp 5529  df-rel 5530  df-cnv 5531  df-dm 5533  df-rn 5534  df-fn 6331  df-fo 6334  df-wdom 9017
This theorem is referenced by:  brwdomi  9020  brwdomn0  9021  0wdom  9022  fowdom  9023  domwdom  9026  wdomnumr  9479
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