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Theorem rnmptss 6878
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 6866 . 2 (∀𝑥𝐴 𝐵𝐶𝐹:𝐴𝐶)
3 frn 6513 . 2 (𝐹:𝐴𝐶 → ran 𝐹𝐶)
42, 3sylbi 218 1 (∀𝑥𝐴 𝐵𝐶 → ran 𝐹𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  wral 3135  wss 3933  cmpt 5137  ran crn 5549  wf 6344
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356
This theorem is referenced by:  mptexw  7643  iunon  7965  iinon  7966  gruiun  10209  subdrgint  19511  smadiadetlem3lem2  21204  tgiun  21515  ustuqtop0  22776  metustss  23088  efabl  25061  efsubm  25062  fnpreimac  30344  swrdrn2  30555  gsummpt2co  30613  psgnfzto1stlem  30669  locfinreflem  31003  prodindf  31181  gsumesum  31217  esumlub  31218  esumgect  31248  esum2d  31251  ldgenpisyslem1  31321  sxbrsigalem0  31428  omscl  31452  omsmon  31455  carsgclctunlem2  31476  carsgclctunlem3  31477  pmeasadd  31482  hgt750lemb  31826  mnurndlem2  40495  suprnmpt  41306  rnmptssrn  41318  wessf1ornlem  41321  rnmptssd  41334  rnmptssbi  41410  liminflelimsuplem  41932  fourierdlem31  42300  fourierdlem53  42321  fourierdlem111  42379  ioorrnopnlem  42466  saliuncl  42484  salexct3  42502  salgensscntex  42504  sge0rnre  42523  sge0tsms  42539  sge0cl  42540  sge0fsum  42546  sge0sup  42550  sge0gerp  42554  sge0pnffigt  42555  sge0lefi  42557  sge0xaddlem1  42592  sge0xaddlem2  42593  meadjiunlem  42624  meadjiun  42625
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