Theorem rnmptss 7064

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 7051	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ↔ 𝐹:𝐴⟶𝐶)
3		frn 6662	. 2 ⊢ (𝐹:𝐴⟶𝐶 → ran 𝐹 ⊆ 𝐶)
4	2, 3	sylbi 218	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ran 𝐹 ⊆ 𝐶)

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119  ∀wral 3053   ⊆ wss 3883   ↦ cmpt 5153  ran crn 5619  ⟶wf 6481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-pr 5362

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-sn 4556  df-pr 4558  df-op 4562  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-fun 6487  df-fn 6488  df-f 6489

This theorem is referenced by:  rnmptssd  7065  mptexw  7895  iunon  8269  iinon  8270  gruiun  10713  subdrgint  20775  smadiadetlem3lem2  22650  tgiun  22962  ustuqtop0  24223  metustss  24534  efabl  26532  efsubm  26533  fnpreimac  32762  prodindf  32941  swrdrn2  33033  gsummpt2co  33129  psgnfzto1stlem  33181  elrgspnsubrunlem1  33328  nsgmgc  33495  nsgqusf1olem1  33496  algextdeglem2  33902  algextdeglem4  33904  locfinreflem  34024  rspectopn  34051  zarcls  34058  zartopn  34059  gsumesum  34243  esumlub  34244  esumgect  34274  esum2d  34277  ldgenpisyslem1  34347  sxbrsigalem0  34455  omscl  34479  omsmon  34482  carsgclctunlem2  34503  carsgclctunlem3  34504  pmeasadd  34509  hgt750lemb  34840  mnurndlem2  44726  suprnmpt  45621  rnmptssrn  45629  wessf1ornlem  45632  rnmptssbi  45704  liminflelimsuplem  46218  fourierdlem53  46602  fourierdlem111  46660  ioorrnopnlem  46747  salexct3  46785  salgensscntex  46787  sge0rnre  46807  sge0tsms  46823  sge0cl  46824  sge0fsum  46830  sge0sup  46834  sge0gerp  46838  sge0pnffigt  46839  sge0lefi  46841  sge0xaddlem1  46876  sge0xaddlem2  46877  meadjiunlem  46908  sinnpoly  47354