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Theorem rnmptss 6879
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 6867 . 2 (∀𝑥𝐴 𝐵𝐶𝐹:𝐴𝐶)
3 frn 6514 . 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 3138  wss 3935  cmpt 5138  ran crn 5550  wf 6345
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 2793  ax-sep 5195  ax-nul 5202  ax-pr 5321
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 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4833  df-br 5059  df-opab 5121  df-mpt 5139  df-id 5454  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-rn 5560  df-res 5561  df-ima 5562  df-iota 6308  df-fun 6351  df-fn 6352  df-f 6353  df-fv 6357
This theorem is referenced by:  mptexw  7645  iunon  7967  iinon  7968  gruiun  10210  subdrgint  19513  smadiadetlem3lem2  21206  tgiun  21517  ustuqtop0  22778  metustss  23090  efabl  25061  efsubm  25062  fnpreimac  30345  swrdrn2  30556  gsummpt2co  30614  psgnfzto1stlem  30670  locfinreflem  31004  prodindf  31182  gsumesum  31218  esumlub  31219  esumgect  31249  esum2d  31252  ldgenpisyslem1  31322  sxbrsigalem0  31429  omscl  31453  omsmon  31456  carsgclctunlem2  31477  carsgclctunlem3  31478  pmeasadd  31483  hgt750lemb  31827  mnurndlem2  40498  suprnmpt  41310  rnmptssrn  41322  wessf1ornlem  41325  rnmptssd  41338  rnmptssbi  41414  liminflelimsuplem  41936  fourierdlem31  42304  fourierdlem53  42325  fourierdlem111  42383  ioorrnopnlem  42470  saliuncl  42488  salexct3  42506  salgensscntex  42508  sge0rnre  42527  sge0tsms  42543  sge0cl  42544  sge0fsum  42550  sge0sup  42554  sge0gerp  42558  sge0pnffigt  42559  sge0lefi  42561  sge0xaddlem1  42596  sge0xaddlem2  42597  meadjiunlem  42628  meadjiun  42629
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