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

Theorem funrnex 7967
Description: If the domain of a function exists, so does its range. Part of Theorem 4.15(v) of [Monk1] p. 46. This theorem is derived using the Axiom of Replacement in the form of funex 7236. (Contributed by NM, 11-Nov-1995.)
Assertion
Ref Expression
funrnex (dom 𝐹𝐵 → (Fun 𝐹 → ran 𝐹 ∈ V))

Proof of Theorem funrnex
StepHypRef Expression
1 funex 7236 . . 3 ((Fun 𝐹 ∧ dom 𝐹𝐵) → 𝐹 ∈ V)
21ex 411 . 2 (Fun 𝐹 → (dom 𝐹𝐵𝐹 ∈ V))
3 rnexg 7915 . 2 (𝐹 ∈ V → ran 𝐹 ∈ V)
42, 3syl6com 37 1 (dom 𝐹𝐵 → (Fun 𝐹 → ran 𝐹 ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2099  Vcvv 3462  dom cdm 5682  ran crn 5683  Fun wfun 6548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562
This theorem is referenced by:  zfrep6  7968  focdmex  7969  tz7.48-3  8474  inf0  9664  axcc2lem  10479  zorn2lem4  10542  fnct  10580  tfsconcatrev  43014
  Copyright terms: Public domain W3C validator