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Theorem brwdomn0 8418
Description: Weak dominance over nonempty sets. (Contributed by Stefan O'Rear, 11-Feb-2015.) (Revised by Mario Carneiro, 5-May-2015.)
Assertion
Ref Expression
brwdomn0 (𝑋 ≠ ∅ → (𝑋* 𝑌 ↔ ∃𝑧 𝑧:𝑌onto𝑋))
Distinct variable groups:   𝑧,𝑋   𝑧,𝑌

Proof of Theorem brwdomn0
StepHypRef Expression
1 relwdom 8415 . . . 4 Rel ≼*
21brrelex2i 5119 . . 3 (𝑋* 𝑌𝑌 ∈ V)
32a1i 11 . 2 (𝑋 ≠ ∅ → (𝑋* 𝑌𝑌 ∈ V))
4 fof 6072 . . . . . 6 (𝑧:𝑌onto𝑋𝑧:𝑌𝑋)
5 fdm 6008 . . . . . 6 (𝑧:𝑌𝑋 → dom 𝑧 = 𝑌)
64, 5syl 17 . . . . 5 (𝑧:𝑌onto𝑋 → dom 𝑧 = 𝑌)
7 vex 3189 . . . . . 6 𝑧 ∈ V
87dmex 7046 . . . . 5 dom 𝑧 ∈ V
96, 8syl6eqelr 2707 . . . 4 (𝑧:𝑌onto𝑋𝑌 ∈ V)
109exlimiv 1855 . . 3 (∃𝑧 𝑧:𝑌onto𝑋𝑌 ∈ V)
1110a1i 11 . 2 (𝑋 ≠ ∅ → (∃𝑧 𝑧:𝑌onto𝑋𝑌 ∈ V))
12 brwdom 8416 . . . 4 (𝑌 ∈ V → (𝑋* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋)))
13 df-ne 2791 . . . . . 6 (𝑋 ≠ ∅ ↔ ¬ 𝑋 = ∅)
14 biorf 420 . . . . . 6 𝑋 = ∅ → (∃𝑧 𝑧:𝑌onto𝑋 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋)))
1513, 14sylbi 207 . . . . 5 (𝑋 ≠ ∅ → (∃𝑧 𝑧:𝑌onto𝑋 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋)))
1615bicomd 213 . . . 4 (𝑋 ≠ ∅ → ((𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋) ↔ ∃𝑧 𝑧:𝑌onto𝑋))
1712, 16sylan9bbr 736 . . 3 ((𝑋 ≠ ∅ ∧ 𝑌 ∈ V) → (𝑋* 𝑌 ↔ ∃𝑧 𝑧:𝑌onto𝑋))
1817ex 450 . 2 (𝑋 ≠ ∅ → (𝑌 ∈ V → (𝑋* 𝑌 ↔ ∃𝑧 𝑧:𝑌onto𝑋)))
193, 11, 18pm5.21ndd 369 1 (𝑋 ≠ ∅ → (𝑋* 𝑌 ↔ ∃𝑧 𝑧:𝑌onto𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383   = wceq 1480  wex 1701  wcel 1987  wne 2790  Vcvv 3186  c0 3891   class class class wbr 4613  dom cdm 5074  wf 5843  ontowfo 5845  * cwdom 8406
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-xp 5080  df-rel 5081  df-cnv 5082  df-dm 5084  df-rn 5085  df-fn 5850  df-f 5851  df-fo 5853  df-wdom 8408
This theorem is referenced by:  brwdom2  8422  wdomtr  8424  wdompwdom  8427  canthwdom  8428  wdomfil  8828  fin1a2lem7  9172
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