Theorem rnmptss 7119

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 7106	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ↔ 𝐹:𝐴⟶𝐶)
3		frn 6714	. 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 1567   ∈ wcel 2149  ∀wral 3085   ⊆ wss 3913   ↦ cmpt 5196  ran crn 5663  ⟶wf 6533

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-fun 6539  df-fn 6540  df-f 6541

This theorem is referenced by:  rnmptssd  7120  mptexw  7949  iunon  8325  iinon  8326  gruiun  10783  subdrgint  20883  smadiadetlem3lem2  22792  tgiun  23104  ustuqtop0  24365  metustss  24676  efabl  26680  efsubm  26681  fnpreimac  32955  prodindf  33122  swrdrn2  33214  gsummpt2co  33308  psgnfzto1stlem  33360  elrgspnsubrunlem1  33507  nsgmgc  33664  nsgqusf1olem1  33665  algextdeglem2  34052  algextdeglem4  34054  locfinreflem  34174  rspectopn  34201  zarcls  34208  zartopn  34209  gsumesum  34393  esumlub  34394  esumgect  34424  esum2d  34427  ldgenpisyslem1  34497  sxbrsigalem0  34605  omscl  34629  omsmon  34632  carsgclctunlem2  34653  carsgclctunlem3  34654  pmeasadd  34659  hgt750lemb  34987  mnurndlem2  44883  suprnmpt  45783  rnmptssrn  45791  wessf1ornlem  45794  rnmptssbi  45866  liminflelimsuplem  46380  fourierdlem53  46764  fourierdlem111  46822  ioorrnopnlem  46909  salexct3  46947  salgensscntex  46949  sge0rnre  46969  sge0tsms  46985  sge0cl  46986  sge0fsum  46992  sge0sup  46996  sge0gerp  47000  sge0pnffigt  47001  sge0lefi  47003  sge0xaddlem1  47038  sge0xaddlem2  47039  meadjiunlem  47070  sinnpoly  47516