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Theorem brwdomn0 8630
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 8627 . . . 4 Rel ≼*
21brrelex2i 5299 . . 3 (𝑋* 𝑌𝑌 ∈ V)
32a1i 11 . 2 (𝑋 ≠ ∅ → (𝑋* 𝑌𝑌 ∈ V))
4 fof 6256 . . . . . 6 (𝑧:𝑌onto𝑋𝑧:𝑌𝑋)
5 fdm 6191 . . . . . 6 (𝑧:𝑌𝑋 → dom 𝑧 = 𝑌)
64, 5syl 17 . . . . 5 (𝑧:𝑌onto𝑋 → dom 𝑧 = 𝑌)
7 vex 3354 . . . . . 6 𝑧 ∈ V
87dmex 7246 . . . . 5 dom 𝑧 ∈ V
96, 8syl6eqelr 2859 . . . 4 (𝑧:𝑌onto𝑋𝑌 ∈ V)
109exlimiv 2010 . . 3 (∃𝑧 𝑧:𝑌onto𝑋𝑌 ∈ V)
1110a1i 11 . 2 (𝑋 ≠ ∅ → (∃𝑧 𝑧:𝑌onto𝑋𝑌 ∈ V))
12 brwdom 8628 . . . 4 (𝑌 ∈ V → (𝑋* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑧 𝑧:𝑌onto𝑋)))
13 df-ne 2944 . . . . . 6 (𝑋 ≠ ∅ ↔ ¬ 𝑋 = ∅)
14 biorf 920 . . . . . 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 500 . . 3 ((𝑋 ≠ ∅ ∧ 𝑌 ∈ V) → (𝑋* 𝑌 ↔ ∃𝑧 𝑧:𝑌onto𝑋))
1817ex 397 . 2 (𝑋 ≠ ∅ → (𝑌 ∈ V → (𝑋* 𝑌 ↔ ∃𝑧 𝑧:𝑌onto𝑋)))
193, 11, 18pm5.21ndd 368 1 (𝑋 ≠ ∅ → (𝑋* 𝑌 ↔ ∃𝑧 𝑧:𝑌onto𝑋))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 834   = wceq 1631  wex 1852  wcel 2145  wne 2943  Vcvv 3351  c0 4063   class class class wbr 4786  dom cdm 5249  wf 6027  ontowfo 6029  * cwdom 8618
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-xp 5255  df-rel 5256  df-cnv 5257  df-dm 5259  df-rn 5260  df-fn 6034  df-f 6035  df-fo 6037  df-wdom 8620
This theorem is referenced by:  brwdom2  8634  wdomtr  8636  wdompwdom  8639  canthwdom  8640  wdomfil  9084  fin1a2lem7  9430
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