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Theorem rnmptss 7143
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 = (𝑥𝐴𝐵)
Assertion
Ref Expression
rnmptss (∀𝑥𝐴 𝐵𝐶 → ran 𝐹𝐶)
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 = (𝑥𝐴𝐵)
21fmpt 7130 . 2 (∀𝑥𝐴 𝐵𝐶𝐹:𝐴𝐶)
3 frn 6743 . 2 (𝐹:𝐴𝐶 → ran 𝐹𝐶)
42, 3sylbi 217 1 (∀𝑥𝐴 𝐵𝐶 → ran 𝐹𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  wral 3061  wss 3951  cmpt 5225  ran crn 5686  wf 6557
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-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
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-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  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-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-mpt 5226  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
This theorem is referenced by:  mptexw  7977  iunon  8379  iinon  8380  gruiun  10839  subdrgint  20804  smadiadetlem3lem2  22673  tgiun  22986  ustuqtop0  24249  metustss  24564  efabl  26592  efsubm  26593  fnpreimac  32681  prodindf  32848  swrdrn2  32939  gsummpt2co  33051  psgnfzto1stlem  33120  elrgspnsubrunlem1  33251  nsgmgc  33440  nsgqusf1olem1  33441  algextdeglem2  33759  algextdeglem4  33761  locfinreflem  33839  rspectopn  33866  zarcls  33873  zartopn  33874  gsumesum  34060  esumlub  34061  esumgect  34091  esum2d  34094  ldgenpisyslem1  34164  sxbrsigalem0  34273  omscl  34297  omsmon  34300  carsgclctunlem2  34321  carsgclctunlem3  34322  pmeasadd  34327  hgt750lemb  34671  mnurndlem2  44301  suprnmpt  45179  rnmptssrn  45187  wessf1ornlem  45190  rnmptssd  45201  rnmptssbi  45267  liminflelimsuplem  45790  fourierdlem53  46174  fourierdlem111  46232  ioorrnopnlem  46319  salexct3  46357  salgensscntex  46359  sge0rnre  46379  sge0tsms  46395  sge0cl  46396  sge0fsum  46402  sge0sup  46406  sge0gerp  46410  sge0pnffigt  46411  sge0lefi  46413  sge0xaddlem1  46448  sge0xaddlem2  46449  meadjiunlem  46480
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