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Theorem domwdom 8639
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 2944 . . . . . . . 8 (𝑋 ≠ ∅ ↔ ¬ 𝑋 = ∅)
21biimpri 218 . . . . . . 7 𝑋 = ∅ → 𝑋 ≠ ∅)
32adantl 467 . . . . . 6 ((𝑋𝑌 ∧ ¬ 𝑋 = ∅) → 𝑋 ≠ ∅)
4 reldom 8119 . . . . . . . . 9 Rel ≼
54brrelexi 5297 . . . . . . . 8 (𝑋𝑌𝑋 ∈ V)
6 0sdomg 8249 . . . . . . . 8 (𝑋 ∈ V → (∅ ≺ 𝑋𝑋 ≠ ∅))
75, 6syl 17 . . . . . . 7 (𝑋𝑌 → (∅ ≺ 𝑋𝑋 ≠ ∅))
87adantr 466 . . . . . 6 ((𝑋𝑌 ∧ ¬ 𝑋 = ∅) → (∅ ≺ 𝑋𝑋 ≠ ∅))
93, 8mpbird 247 . . . . 5 ((𝑋𝑌 ∧ ¬ 𝑋 = ∅) → ∅ ≺ 𝑋)
10 simpl 468 . . . . 5 ((𝑋𝑌 ∧ ¬ 𝑋 = ∅) → 𝑋𝑌)
11 fodomr 8271 . . . . 5 ((∅ ≺ 𝑋𝑋𝑌) → ∃𝑦 𝑦:𝑌onto𝑋)
129, 10, 11syl2anc 573 . . . 4 ((𝑋𝑌 ∧ ¬ 𝑋 = ∅) → ∃𝑦 𝑦:𝑌onto𝑋)
1312ex 397 . . 3 (𝑋𝑌 → (¬ 𝑋 = ∅ → ∃𝑦 𝑦:𝑌onto𝑋))
1413orrd 852 . 2 (𝑋𝑌 → (𝑋 = ∅ ∨ ∃𝑦 𝑦:𝑌onto𝑋))
154brrelex2i 5298 . . 3 (𝑋𝑌𝑌 ∈ V)
16 brwdom 8632 . . 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  wa 382  wo 836   = wceq 1631  wex 1852  wcel 2145  wne 2943  Vcvv 3351  c0 4063   class class class wbr 4787  ontowfo 6028  cdom 8111  csdm 8112  * cwdom 8622
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 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  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 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-er 7900  df-en 8114  df-dom 8115  df-sdom 8116  df-wdom 8624
This theorem is referenced by:  wdomen1  8641  wdomen2  8642  wdom2d  8645  wdomima2g  8651  unxpwdom2  8653  unxpwdom  8654  harwdom  8655  wdomfil  9088  wdomnumr  9091  pwcdadom  9244  hsmexlem1  9454  hsmexlem4  9457
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