Theorem rnmptss 7157

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 7144	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ↔ 𝐹:𝐴⟶𝐶)
3		frn 6754	. 2 ⊢ (𝐹:𝐴⟶𝐶 → ran 𝐹 ⊆ 𝐶)
4	2, 3	sylbi 217	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ran 𝐹 ⊆ 𝐶)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108  ∀wral 3067   ⊆ wss 3976   ↦ cmpt 5249  ran crn 5701  ⟶wf 6569

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-fun 6575  df-fn 6576  df-f 6577

This theorem is referenced by:  mptexw  7993  iunon  8395  iinon  8396  gruiun  10868  subdrgint  20826  smadiadetlem3lem2  22694  tgiun  23007  ustuqtop0  24270  metustss  24585  efabl  26610  efsubm  26611  fnpreimac  32689  swrdrn2  32921  gsummpt2co  33031  psgnfzto1stlem  33093  nsgmgc  33405  nsgqusf1olem1  33406  algextdeglem2  33709  algextdeglem4  33711  locfinreflem  33786  rspectopn  33813  zarcls  33820  zartopn  33821  prodindf  33987  gsumesum  34023  esumlub  34024  esumgect  34054  esum2d  34057  ldgenpisyslem1  34127  sxbrsigalem0  34236  omscl  34260  omsmon  34263  carsgclctunlem2  34284  carsgclctunlem3  34285  pmeasadd  34290  hgt750lemb  34633  mnurndlem2  44251  suprnmpt  45081  rnmptssrn  45089  wessf1ornlem  45092  rnmptssd  45103  rnmptssbi  45170  liminflelimsuplem  45696  fourierdlem53  46080  fourierdlem111  46138  ioorrnopnlem  46225  salexct3  46263  salgensscntex  46265  sge0rnre  46285  sge0tsms  46301  sge0cl  46302  sge0fsum  46308  sge0sup  46312  sge0gerp  46316  sge0pnffigt  46317  sge0lefi  46319  sge0xaddlem1  46354  sge0xaddlem2  46355  meadjiunlem  46386