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Theorem domwdom 9333
Description: Weak dominance is implied by dominance in the usual sense. (Contributed by Stefan O'Rear, 11-Feb-2015.)
Assertion
Ref Expression
domwdom (𝑋𝑌𝑋* 𝑌)

Proof of Theorem domwdom
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 neqne 2951 . . . . . . 7 𝑋 = ∅ → 𝑋 ≠ ∅)
21adantl 482 . . . . . 6 ((𝑋𝑌 ∧ ¬ 𝑋 = ∅) → 𝑋 ≠ ∅)
3 reldom 8739 . . . . . . . . 9 Rel ≼
43brrelex1i 5643 . . . . . . . 8 (𝑋𝑌𝑋 ∈ V)
5 0sdomg 8891 . . . . . . . 8 (𝑋 ∈ V → (∅ ≺ 𝑋𝑋 ≠ ∅))
64, 5syl 17 . . . . . . 7 (𝑋𝑌 → (∅ ≺ 𝑋𝑋 ≠ ∅))
76adantr 481 . . . . . 6 ((𝑋𝑌 ∧ ¬ 𝑋 = ∅) → (∅ ≺ 𝑋𝑋 ≠ ∅))
82, 7mpbird 256 . . . . 5 ((𝑋𝑌 ∧ ¬ 𝑋 = ∅) → ∅ ≺ 𝑋)
9 simpl 483 . . . . 5 ((𝑋𝑌 ∧ ¬ 𝑋 = ∅) → 𝑋𝑌)
10 fodomr 8915 . . . . 5 ((∅ ≺ 𝑋𝑋𝑌) → ∃𝑦 𝑦:𝑌onto𝑋)
118, 9, 10syl2anc 584 . . . 4 ((𝑋𝑌 ∧ ¬ 𝑋 = ∅) → ∃𝑦 𝑦:𝑌onto𝑋)
1211ex 413 . . 3 (𝑋𝑌 → (¬ 𝑋 = ∅ → ∃𝑦 𝑦:𝑌onto𝑋))
1312orrd 860 . 2 (𝑋𝑌 → (𝑋 = ∅ ∨ ∃𝑦 𝑦:𝑌onto𝑋))
143brrelex2i 5644 . . 3 (𝑋𝑌𝑌 ∈ V)
15 brwdom 9326 . . 3 (𝑌 ∈ V → (𝑋* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑦 𝑦:𝑌onto𝑋)))
1614, 15syl 17 . 2 (𝑋𝑌 → (𝑋* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑦 𝑦:𝑌onto𝑋)))
1713, 16mpbird 256 1 (𝑋𝑌𝑋* 𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  wo 844   = wceq 1539  wex 1782  wcel 2106  wne 2943  Vcvv 3432  c0 4256   class class class wbr 5074  ontowfo 6431  cdom 8731  csdm 8732  * cwdom 9323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-en 8734  df-dom 8735  df-sdom 8736  df-wdom 9324
This theorem is referenced by:  wdomen1  9335  wdomen2  9336  wdom2d  9339  wdomima2g  9345  unxpwdom2  9347  unxpwdom  9348  harwdom  9350  wdomfil  9817  wdomnumr  9820  pwdjudom  9972  hsmexlem1  10182  hsmexlem4  10185
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