ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  rnmptss Unicode version

Theorem rnmptss 5628
Description: The range of an operation given by the maps-to notation as a subset. (Contributed by Thierry Arnoux, 24-Sep-2017.)
Hypothesis
Ref Expression
rnmptss.1  |-  F  =  ( x  e.  A  |->  B )
Assertion
Ref Expression
rnmptss  |-  ( A. x  e.  A  B  e.  C  ->  ran  F  C_  C )
Distinct variable groups:    x, A    x, C
Allowed substitution hints:    B( x)    F( x)

Proof of Theorem rnmptss
StepHypRef Expression
1 rnmptss.1 . . 3  |-  F  =  ( x  e.  A  |->  B )
21fmpt 5617 . 2  |-  ( A. x  e.  A  B  e.  C  <->  F : A --> C )
3 frn 5328 . 2  |-  ( F : A --> C  ->  ran  F  C_  C )
42, 3sylbi 120 1  |-  ( A. x  e.  A  B  e.  C  ->  ran  F  C_  C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1335    e. wcel 2128   A.wral 2435    C_ wss 3102    |-> cmpt 4025   ran crn 4587   -->wf 5166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-fv 5178
This theorem is referenced by:  tgiun  12484  dvrecap  13088
  Copyright terms: Public domain W3C validator