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Theorem rnmptss 6050
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 6042 . 2  |-  ( A. x  e.  A  B  e.  C  <->  F : A --> C )
3 frn 5737 . 2  |-  ( F : A --> C  ->  ran  F  C_  C )
42, 3sylbi 195 1  |-  ( A. x  e.  A  B  e.  C  ->  ran  F  C_  C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   A.wral 2814    C_ wss 3476    |-> cmpt 4505   ran crn 5000   -->wf 5584
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-fv 5596
This theorem is referenced by:  smadiadetlem3lem2  18964  ustuqtop0  20506  metustssOLD  20819  metustss  20820  gsummpt2co  27462  gsumesum  27735  esumlub  27736  sxbrsigalem0  27910  omsmon  27935  suprnmpt  31057  fourierdlem31  31466  fourierdlem53  31488  fourierdlem111  31546
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