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Theorem rnmptss 6036
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 6028 . 2  |-  ( A. x  e.  A  B  e.  C  <->  F : A --> C )
3 frn 5719 . 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 1398    e. wcel 1823   A.wral 2804    C_ wss 3461    |-> cmpt 4497   ran crn 4989   -->wf 5566
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-fv 5578
This theorem is referenced by:  iunon  7001  iinon  7003  gruiun  9166  smadiadetlem3lem2  19336  tgiun  19648  ustuqtop0  20909  metustssOLD  21222  metustss  21223  efabl  23103  efsubm  23104  gsummpt2co  28005  locfinreflem  28078  gsumesum  28288  esumlub  28289  esumgect  28319  esum2d  28322  sxbrsigalem0  28479  omscl  28503  omsmon  28506  carsgclctunlem2  28527  carsgclctunlem3  28528  suprnmpt  31691  fourierdlem31  32159  fourierdlem53  32181  fourierdlem111  32239
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