Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  rnmptss Structured version   Visualization version   GIF version

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 6513 . 2 (𝐹:𝐴𝐶 → ran 𝐹𝐶)
42, 3sylbi 219 1 (∀𝑥𝐴 𝐵𝐶 → ran 𝐹𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1531   ∈ wcel 2108  ∀wral 3136   ⊆ wss 3934   ↦ 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 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  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  7646  iunon  7968  iinon  7969  gruiun  10213  subdrgint  19574  smadiadetlem3lem2  21268  tgiun  21579  ustuqtop0  22841  metustss  23153  efabl  25126  efsubm  25127  fnpreimac  30408  swrdrn2  30621  gsummpt2co  30679  psgnfzto1stlem  30735  locfinreflem  31097  prodindf  31275  gsumesum  31311  esumlub  31312  esumgect  31342  esum2d  31345  ldgenpisyslem1  31415  sxbrsigalem0  31522  omscl  31546  omsmon  31549  carsgclctunlem2  31570  carsgclctunlem3  31571  pmeasadd  31576  hgt750lemb  31920  mnurndlem2  40608  suprnmpt  41419  rnmptssrn  41431  wessf1ornlem  41434  rnmptssd  41447  rnmptssbi  41523  liminflelimsuplem  42045  fourierdlem31  42413  fourierdlem53  42434  fourierdlem111  42492  ioorrnopnlem  42579  saliuncl  42597  salexct3  42615  salgensscntex  42617  sge0rnre  42636  sge0tsms  42652  sge0cl  42653  sge0fsum  42659  sge0sup  42663  sge0gerp  42667  sge0pnffigt  42668  sge0lefi  42670  sge0xaddlem1  42705  sge0xaddlem2  42706  meadjiunlem  42737  meadjiun  42738
 Copyright terms: Public domain W3C validator