Theorem rnmptss 6392

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 = ( x e. A |-> B )

Assertion

Ref	Expression
rnmptss	|- ( A. x e. A B e. C -> ran F C_ C )

Distinct variable groups:   x, A   x, C

Allowed substitution hints:   B( x)   F( x)

Proof of Theorem rnmptss

Step	Hyp	Ref	Expression
1		rnmptss.1	. . 3 |- F = ( x e. A |-> B )
2	1	fmpt 6381	. 2 |- ( A. x e. A B e. C <-> F : A --> C )
3		frn 6053	. 2 |- ( F : A --> C -> ran F C_ C )
4	2, 3	sylbi 207	1 |- ( A. x e. A B e. C -> ran F C_ C )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   |-> cmpt 4729   ran crn 5115   -->wf 5884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896

This theorem is referenced by:  iunon  7436  iinon  7437  gruiun  9621  smadiadetlem3lem2  20473  tgiun  20783  ustuqtop0  22044  metustss  22356  efabl  24296  efsubm  24297  gsummpt2co  29780  psgnfzto1stlem  29850  locfinreflem  29907  prodindf  30085  gsumesum  30121  esumlub  30122  esumgect  30152  esum2d  30155  ldgenpisyslem1  30226  sxbrsigalem0  30333  omscl  30357  omsmon  30360  carsgclctunlem2  30381  carsgclctunlem3  30382  pmeasadd  30387  hgt750lemb  30734  suprnmpt  39355  rnmptssrn  39368  wessf1ornlem  39371  rnmptssd  39385  rnmptssbi  39477  liminflelimsuplem  40007  fourierdlem31  40355  fourierdlem53  40376  fourierdlem111  40434  ioorrnopnlem  40524  saliuncl  40542  salexct3  40560  salgensscntex  40562  sge0rnre  40581  sge0tsms  40597  sge0cl  40598  sge0fsum  40604  sge0sup  40608  sge0gerp  40612  sge0pnffigt  40613  sge0lefi  40615  sge0xaddlem1  40650  sge0xaddlem2  40651  meadjiunlem  40682  meadjiun  40683