MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  wdomref Structured version   Visualization version   GIF version

Theorem wdomref 9612
Description: Reflexivity of weak dominance. (Contributed by Stefan O'Rear, 11-Feb-2015.)
Assertion
Ref Expression
wdomref (𝑋𝑉𝑋* 𝑋)

Proof of Theorem wdomref
StepHypRef Expression
1 resiexg 7934 . 2 (𝑋𝑉 → ( I ↾ 𝑋) ∈ V)
2 f1oi 6886 . . 3 ( I ↾ 𝑋):𝑋1-1-onto𝑋
3 f1ofo 6855 . . 3 (( I ↾ 𝑋):𝑋1-1-onto𝑋 → ( I ↾ 𝑋):𝑋onto𝑋)
42, 3ax-mp 5 . 2 ( I ↾ 𝑋):𝑋onto𝑋
5 fowdom 9611 . 2 ((( I ↾ 𝑋) ∈ V ∧ ( I ↾ 𝑋):𝑋onto𝑋) → 𝑋* 𝑋)
61, 4, 5sylancl 586 1 (𝑋𝑉𝑋* 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2108  Vcvv 3480   class class class wbr 5143   I cid 5577  cres 5687  ontowfo 6559  1-1-ontowf1o 6560  * cwdom 9604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-wdom 9605
This theorem is referenced by:  hsmexlem3  10468  hsmexlem5  10470
  Copyright terms: Public domain W3C validator