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Theorem rnmptss 6358
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 6347 . 2 (∀𝑥𝐴 𝐵𝐶𝐹:𝐴𝐶)
3 frn 6020 . 2 (𝐹:𝐴𝐶 → ran 𝐹𝐶)
42, 3sylbi 207 1 (∀𝑥𝐴 𝐵𝐶 → ran 𝐹𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  wral 2908  wss 3560  cmpt 4683  ran crn 5085  wf 5853
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-fv 5865
This theorem is referenced by:  iunon  7396  iinon  7397  gruiun  9581  smadiadetlem3lem2  20413  tgiun  20723  ustuqtop0  21984  metustss  22296  efabl  24234  efsubm  24235  gsummpt2co  29607  psgnfzto1stlem  29677  locfinreflem  29731  gsumesum  29944  esumlub  29945  esumgect  29975  esum2d  29978  ldgenpisyslem1  30049  sxbrsigalem0  30156  omscl  30180  omsmon  30183  carsgclctunlem2  30204  carsgclctunlem3  30205  pmeasadd  30210  suprnmpt  38864  rnmptssrn  38877  wessf1ornlem  38880  rnmptssd  38894  fourierdlem31  39692  fourierdlem53  39713  fourierdlem111  39771  ioorrnopnlem  39861  saliuncl  39879  salexct3  39897  salgensscntex  39899  sge0rnre  39918  sge0tsms  39934  sge0cl  39935  sge0fsum  39941  sge0sup  39945  sge0gerp  39949  sge0pnffigt  39950  sge0lefi  39952  sge0xaddlem1  39987  sge0xaddlem2  39988  meadjiunlem  40019  meadjiun  40020
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