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Theorem brwdom 8469
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 3210 . 2 (𝑌𝑉𝑌 ∈ V)
2 relwdom 8468 . . . . 5 Rel ≼*
32brrelexi 5156 . . . 4 (𝑋* 𝑌𝑋 ∈ V)
43a1i 11 . . 3 (𝑌 ∈ V → (𝑋* 𝑌𝑋 ∈ V))
5 0ex 4788 . . . . . 6 ∅ ∈ V
6 eleq1a 2695 . . . . . 6 (∅ ∈ V → (𝑋 = ∅ → 𝑋 ∈ V))
75, 6ax-mp 5 . . . . 5 (𝑋 = ∅ → 𝑋 ∈ V)
8 forn 6116 . . . . . . 7 (𝑧:𝑌onto𝑋 → ran 𝑧 = 𝑋)
9 vex 3201 . . . . . . . 8 𝑧 ∈ V
109rnex 7097 . . . . . . 7 ran 𝑧 ∈ V
118, 10syl6eqelr 2709 . . . . . 6 (𝑧:𝑌onto𝑋𝑋 ∈ V)
1211exlimiv 1857 . . . . 5 (∃𝑧 𝑧:𝑌onto𝑋𝑋 ∈ V)
137, 12jaoi 394 . . . 4 ((𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋) → 𝑋 ∈ V)
1413a1i 11 . . 3 (𝑌 ∈ V → ((𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋) → 𝑋 ∈ V))
15 eqeq1 2625 . . . . . 6 (𝑥 = 𝑋 → (𝑥 = ∅ ↔ 𝑋 = ∅))
16 foeq3 6111 . . . . . . 7 (𝑥 = 𝑋 → (𝑧:𝑦onto𝑥𝑧:𝑦onto𝑋))
1716exbidv 1849 . . . . . 6 (𝑥 = 𝑋 → (∃𝑧 𝑧:𝑦onto𝑥 ↔ ∃𝑧 𝑧:𝑦onto𝑋))
1815, 17orbi12d 746 . . . . 5 (𝑥 = 𝑋 → ((𝑥 = ∅ ∨ ∃𝑧 𝑧:𝑦onto𝑥) ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑦onto𝑋)))
19 foeq2 6110 . . . . . . 7 (𝑦 = 𝑌 → (𝑧:𝑦onto𝑋𝑧:𝑌onto𝑋))
2019exbidv 1849 . . . . . 6 (𝑦 = 𝑌 → (∃𝑧 𝑧:𝑦onto𝑋 ↔ ∃𝑧 𝑧:𝑌onto𝑋))
2120orbi2d 738 . . . . 5 (𝑦 = 𝑌 → ((𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑦onto𝑋) ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋)))
22 df-wdom 8461 . . . . 5 * = {⟨𝑥, 𝑦⟩ ∣ (𝑥 = ∅ ∨ ∃𝑧 𝑧:𝑦onto𝑥)}
2318, 21, 22brabg 4992 . . . 4 ((𝑋 ∈ V ∧ 𝑌 ∈ V) → (𝑋* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋)))
2423expcom 451 . . 3 (𝑌 ∈ V → (𝑋 ∈ V → (𝑋* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋))))
254, 14, 24pm5.21ndd 369 . 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 383   = wceq 1482  wex 1703  wcel 1989  Vcvv 3198  c0 3913   class class class wbr 4651  ran crn 5113  ontowfo 5884  * cwdom 8459
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-xp 5118  df-rel 5119  df-cnv 5120  df-dm 5122  df-rn 5123  df-fn 5889  df-fo 5892  df-wdom 8461
This theorem is referenced by:  brwdomi  8470  brwdomn0  8471  0wdom  8472  fowdom  8473  domwdom  8476  wdomnumr  8884
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