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Theorem domwdom 8439
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 2791 . . . . . . . 8 (𝑋 ≠ ∅ ↔ ¬ 𝑋 = ∅)
21biimpri 218 . . . . . . 7 𝑋 = ∅ → 𝑋 ≠ ∅)
32adantl 482 . . . . . 6 ((𝑋𝑌 ∧ ¬ 𝑋 = ∅) → 𝑋 ≠ ∅)
4 reldom 7921 . . . . . . . . 9 Rel ≼
54brrelexi 5128 . . . . . . . 8 (𝑋𝑌𝑋 ∈ V)
6 0sdomg 8049 . . . . . . . 8 (𝑋 ∈ V → (∅ ≺ 𝑋𝑋 ≠ ∅))
75, 6syl 17 . . . . . . 7 (𝑋𝑌 → (∅ ≺ 𝑋𝑋 ≠ ∅))
87adantr 481 . . . . . 6 ((𝑋𝑌 ∧ ¬ 𝑋 = ∅) → (∅ ≺ 𝑋𝑋 ≠ ∅))
93, 8mpbird 247 . . . . 5 ((𝑋𝑌 ∧ ¬ 𝑋 = ∅) → ∅ ≺ 𝑋)
10 simpl 473 . . . . 5 ((𝑋𝑌 ∧ ¬ 𝑋 = ∅) → 𝑋𝑌)
11 fodomr 8071 . . . . 5 ((∅ ≺ 𝑋𝑋𝑌) → ∃𝑦 𝑦:𝑌onto𝑋)
129, 10, 11syl2anc 692 . . . 4 ((𝑋𝑌 ∧ ¬ 𝑋 = ∅) → ∃𝑦 𝑦:𝑌onto𝑋)
1312ex 450 . . 3 (𝑋𝑌 → (¬ 𝑋 = ∅ → ∃𝑦 𝑦:𝑌onto𝑋))
1413orrd 393 . 2 (𝑋𝑌 → (𝑋 = ∅ ∨ ∃𝑦 𝑦:𝑌onto𝑋))
154brrelex2i 5129 . . 3 (𝑋𝑌𝑌 ∈ V)
16 brwdom 8432 . . 3 (𝑌 ∈ V → (𝑋* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑦 𝑦:𝑌onto𝑋)))
1715, 16syl 17 . 2 (𝑋𝑌 → (𝑋* 𝑌 ↔ (𝑋 = ∅ ∨ ∃𝑦 𝑦:𝑌onto𝑋)))
1814, 17mpbird 247 1 (𝑋𝑌𝑋* 𝑌)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384   = wceq 1480  wex 1701  wcel 1987  wne 2790  Vcvv 3190  c0 3897   class class class wbr 4623  ontowfo 5855  cdom 7913  csdm 7914  * cwdom 8422
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 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
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 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-er 7702  df-en 7916  df-dom 7917  df-sdom 7918  df-wdom 8424
This theorem is referenced by:  wdomen1  8441  wdomen2  8442  wdom2d  8445  wdomima2g  8451  unxpwdom2  8453  unxpwdom  8454  harwdom  8455  wdomfil  8844  wdomnumr  8847  pwcdadom  8998  hsmexlem1  9208  hsmexlem4  9211
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