Theorem rnmptss 7104

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 7091	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ↔ 𝐹:𝐴⟶𝐶)
3		frn 6699	. 2 ⊢ (𝐹:𝐴⟶𝐶 → ran 𝐹 ⊆ 𝐶)
4	2, 3	sylbi 219	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ran 𝐹 ⊆ 𝐶)

Syntax hints:   → wi 4   = wceq 1560   ∈ wcel 2142  ∀wral 3076   ⊆ wss 3904   ↦ cmpt 5181  ran crn 5648  ⟶wf 6517

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-fun 6523  df-fn 6524  df-f 6525

This theorem is referenced by:  rnmptssd  7105  mptexw  7934  iunon  8310  iinon  8311  gruiun  10757  subdrgint  20849  smadiadetlem3lem2  22724  tgiun  23036  ustuqtop0  24297  metustss  24608  efabl  26612  efsubm  26613  fnpreimac  32869  prodindf  33037  swrdrn2  33129  gsummpt2co  33225  psgnfzto1stlem  33277  elrgspnsubrunlem1  33425  nsgmgc  33595  nsgqusf1olem1  33596  algextdeglem2  34012  algextdeglem4  34014  locfinreflem  34134  rspectopn  34161  zarcls  34168  zartopn  34169  gsumesum  34353  esumlub  34354  esumgect  34384  esum2d  34387  ldgenpisyslem1  34457  sxbrsigalem0  34565  omscl  34589  omsmon  34592  carsgclctunlem2  34613  carsgclctunlem3  34614  pmeasadd  34619  hgt750lemb  34947  mnurndlem2  44855  suprnmpt  45749  rnmptssrn  45757  wessf1ornlem  45760  rnmptssbi  45832  liminflelimsuplem  46346  fourierdlem53  46730  fourierdlem111  46788  ioorrnopnlem  46875  salexct3  46913  salgensscntex  46915  sge0rnre  46935  sge0tsms  46951  sge0cl  46952  sge0fsum  46958  sge0sup  46962  sge0gerp  46966  sge0pnffigt  46967  sge0lefi  46969  sge0xaddlem1  47004  sge0xaddlem2  47005  meadjiunlem  47036  sinnpoly  47482