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Theorem domwdom 8748
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 df-ne 3000 . . . . . . . 8 (𝑋 ≠ ∅ ↔ ¬ 𝑋 = ∅)
21biimpri 220 . . . . . . 7 𝑋 = ∅ → 𝑋 ≠ ∅)
32adantl 475 . . . . . 6 ((𝑋𝑌 ∧ ¬ 𝑋 = ∅) → 𝑋 ≠ ∅)
4 reldom 8228 . . . . . . . . 9 Rel ≼
54brrelex1i 5393 . . . . . . . 8 (𝑋𝑌𝑋 ∈ V)
6 0sdomg 8358 . . . . . . . 8 (𝑋 ∈ V → (∅ ≺ 𝑋𝑋 ≠ ∅))
75, 6syl 17 . . . . . . 7 (𝑋𝑌 → (∅ ≺ 𝑋𝑋 ≠ ∅))
87adantr 474 . . . . . 6 ((𝑋𝑌 ∧ ¬ 𝑋 = ∅) → (∅ ≺ 𝑋𝑋 ≠ ∅))
93, 8mpbird 249 . . . . 5 ((𝑋𝑌 ∧ ¬ 𝑋 = ∅) → ∅ ≺ 𝑋)
10 simpl 476 . . . . 5 ((𝑋𝑌 ∧ ¬ 𝑋 = ∅) → 𝑋𝑌)
11 fodomr 8380 . . . . 5 ((∅ ≺ 𝑋𝑋𝑌) → ∃𝑦 𝑦:𝑌onto𝑋)
129, 10, 11syl2anc 579 . . . 4 ((𝑋𝑌 ∧ ¬ 𝑋 = ∅) → ∃𝑦 𝑦:𝑌onto𝑋)
1312ex 403 . . 3 (𝑋𝑌 → (¬ 𝑋 = ∅ → ∃𝑦 𝑦:𝑌onto𝑋))
1413orrd 894 . 2 (𝑋𝑌 → (𝑋 = ∅ ∨ ∃𝑦 𝑦:𝑌onto𝑋))
154brrelex2i 5394 . . 3 (𝑋𝑌𝑌 ∈ V)
16 brwdom 8741 . . 3 (𝑌 ∈ V → (𝑋* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑦 𝑦:𝑌onto𝑋)))
1715, 16syl 17 . 2 (𝑋𝑌 → (𝑋* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑦 𝑦:𝑌onto𝑋)))
1814, 17mpbird 249 1 (𝑋𝑌𝑋* 𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 198  wa 386  wo 878   = wceq 1656  wex 1878  wcel 2164  wne 2999  Vcvv 3414  c0 4144   class class class wbr 4873  ontowfo 6121  cdom 8220  csdm 8221  * cwdom 8731
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-8 2166  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pow 5065  ax-pr 5127  ax-un 7209
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-pw 4380  df-sn 4398  df-pr 4400  df-op 4404  df-uni 4659  df-br 4874  df-opab 4936  df-mpt 4953  df-id 5250  df-xp 5348  df-rel 5349  df-cnv 5350  df-co 5351  df-dm 5352  df-rn 5353  df-res 5354  df-ima 5355  df-fun 6125  df-fn 6126  df-f 6127  df-f1 6128  df-fo 6129  df-f1o 6130  df-er 8009  df-en 8223  df-dom 8224  df-sdom 8225  df-wdom 8733
This theorem is referenced by:  wdomen1  8750  wdomen2  8751  wdom2d  8754  wdomima2g  8760  unxpwdom2  8762  unxpwdom  8763  harwdom  8764  wdomfil  9197  wdomnumr  9200  pwcdadom  9353  hsmexlem1  9563  hsmexlem4  9566
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