Theorem rnmptss 7056

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 7043	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ↔ 𝐹:𝐴⟶𝐶)
3		frn 6658	. 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 1541   ∈ wcel 2111  ∀wral 3047   ⊆ wss 3897   ↦ cmpt 5170  ran crn 5615  ⟶wf 6477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-fun 6483  df-fn 6484  df-f 6485

This theorem is referenced by:  mptexw  7885  iunon  8259  iinon  8260  gruiun  10690  subdrgint  20718  smadiadetlem3lem2  22582  tgiun  22894  ustuqtop0  24155  metustss  24466  efabl  26486  efsubm  26487  fnpreimac  32653  prodindf  32844  swrdrn2  32935  gsummpt2co  33028  psgnfzto1stlem  33069  elrgspnsubrunlem1  33214  nsgmgc  33377  nsgqusf1olem1  33378  algextdeglem2  33731  algextdeglem4  33733  locfinreflem  33853  rspectopn  33880  zarcls  33887  zartopn  33888  gsumesum  34072  esumlub  34073  esumgect  34103  esum2d  34106  ldgenpisyslem1  34176  sxbrsigalem0  34284  omscl  34308  omsmon  34311  carsgclctunlem2  34332  carsgclctunlem3  34333  pmeasadd  34338  hgt750lemb  34669  mnurndlem2  44323  suprnmpt  45219  rnmptssrn  45227  wessf1ornlem  45230  rnmptssd  45241  rnmptssbi  45305  liminflelimsuplem  45821  fourierdlem53  46205  fourierdlem111  46263  ioorrnopnlem  46350  salexct3  46388  salgensscntex  46390  sge0rnre  46410  sge0tsms  46426  sge0cl  46427  sge0fsum  46433  sge0sup  46437  sge0gerp  46441  sge0pnffigt  46442  sge0lefi  46444  sge0xaddlem1  46479  sge0xaddlem2  46480  meadjiunlem  46511  sinnpoly  46930