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Theorem rnmptss 6534
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 6523 . 2 (∀𝑥𝐴 𝐵𝐶𝐹:𝐴𝐶)
3 frn 6193 . 2 (𝐹:𝐴𝐶 → ran 𝐹𝐶)
42, 3sylbi 207 1 (∀𝑥𝐴 𝐵𝐶 → ran 𝐹𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  wral 3061  wss 3723  cmpt 4863  ran crn 5250  wf 6027
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039
This theorem is referenced by:  iunon  7589  iinon  7590  gruiun  9823  smadiadetlem3lem2  20692  tgiun  21004  ustuqtop0  22264  metustss  22576  efabl  24517  efsubm  24518  gsummpt2co  30120  psgnfzto1stlem  30190  locfinreflem  30247  prodindf  30425  gsumesum  30461  esumlub  30462  esumgect  30492  esum2d  30495  ldgenpisyslem1  30566  sxbrsigalem0  30673  omscl  30697  omsmon  30700  carsgclctunlem2  30721  carsgclctunlem3  30722  pmeasadd  30727  hgt750lemb  31074  suprnmpt  39875  rnmptssrn  39888  wessf1ornlem  39891  rnmptssd  39905  rnmptssbi  39995  liminflelimsuplem  40525  fourierdlem31  40872  fourierdlem53  40893  fourierdlem111  40951  ioorrnopnlem  41041  saliuncl  41059  salexct3  41077  salgensscntex  41079  sge0rnre  41098  sge0tsms  41114  sge0cl  41115  sge0fsum  41121  sge0sup  41125  sge0gerp  41129  sge0pnffigt  41130  sge0lefi  41132  sge0xaddlem1  41167  sge0xaddlem2  41168  meadjiunlem  41199  meadjiun  41200
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