Theorem rnmptss 7118

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 ∈ 𝐴 ↦ 𝐵)

Assertion

Ref	Expression
rnmptss	⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ran 𝐹 ⊆ 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 ∈ 𝐴 ↦ 𝐵)
2	1	fmpt 7105	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ↔ 𝐹:𝐴⟶𝐶)
3		frn 6713	. 2 ⊢ (𝐹:𝐴⟶𝐶 → ran 𝐹 ⊆ 𝐶)
4	2, 3	sylbi 220	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ran 𝐹 ⊆ 𝐶)

Syntax hints:   → wi 4   = wceq 1570   ∈ wcel 2143  ∀wral 3079   ⊆ wss 3905   ↦ cmpt 5192  ran crn 5662  ⟶wf 6532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-fun 6538  df-fn 6539  df-f 6540

This theorem is referenced by:  rnmptssd  7119  mptexw  7946  iunon  8322  iinon  8323  gruiun  10779  subdrgint  20906  smadiadetlem3lem2  22824  tgiun  23136  ustuqtop0  24397  metustss  24708  efabl  26715  efsubm  26716  fnpreimac  33015  prodindf  33182  swrdrn2  33274  gsummpt2co  33368  psgnfzto1stlem  33420  elrgspnsubrunlem1  33567  nsgmgc  33721  nsgqusf1olem1  33722  algextdeglem2  34108  algextdeglem4  34110  locfinreflem  34230  rspectopn  34257  zarcls  34264  zartopn  34265  gsumesum  34449  esumlub  34450  esumgect  34480  esum2d  34483  ldgenpisyslem1  34553  sxbrsigalem0  34661  omscl  34685  omsmon  34688  carsgclctunlem2  34709  carsgclctunlem3  34710  pmeasadd  34715  hgt750lemb  35043  mnurndlem2  44992  suprnmpt  45892  rnmptssrn  45900  wessf1ornlem  45903  rnmptssbi  45975  liminflelimsuplem  46489  fourierdlem53  46873  fourierdlem111  46931  ioorrnopnlem  47018  salexct3  47056  salgensscntex  47058  sge0rnre  47078  sge0tsms  47094  sge0cl  47095  sge0fsum  47101  sge0sup  47105  sge0gerp  47109  sge0pnffigt  47110  sge0lefi  47112  sge0xaddlem1  47147  sge0xaddlem2  47148  meadjiunlem  47179  sinnpoly  47628