Theorem rnmptss 7077

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 7064	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ↔ 𝐹:𝐴⟶𝐶)
3		frn 6677	. 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 1540   ∈ wcel 2109  ∀wral 3044   ⊆ wss 3911   ↦ cmpt 5183  ran crn 5632  ⟶wf 6495

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-fun 6501  df-fn 6502  df-f 6503

This theorem is referenced by:  mptexw  7911  iunon  8285  iinon  8286  gruiun  10728  subdrgint  20688  smadiadetlem3lem2  22530  tgiun  22842  ustuqtop0  24104  metustss  24415  efabl  26435  efsubm  26436  fnpreimac  32568  prodindf  32759  swrdrn2  32849  gsummpt2co  32961  psgnfzto1stlem  33030  elrgspnsubrunlem1  33171  nsgmgc  33356  nsgqusf1olem1  33357  algextdeglem2  33681  algextdeglem4  33683  locfinreflem  33803  rspectopn  33830  zarcls  33837  zartopn  33838  gsumesum  34022  esumlub  34023  esumgect  34053  esum2d  34056  ldgenpisyslem1  34126  sxbrsigalem0  34235  omscl  34259  omsmon  34262  carsgclctunlem2  34283  carsgclctunlem3  34284  pmeasadd  34289  hgt750lemb  34620  mnurndlem2  44244  suprnmpt  45141  rnmptssrn  45149  wessf1ornlem  45152  rnmptssd  45163  rnmptssbi  45227  liminflelimsuplem  45746  fourierdlem53  46130  fourierdlem111  46188  ioorrnopnlem  46275  salexct3  46313  salgensscntex  46315  sge0rnre  46335  sge0tsms  46351  sge0cl  46352  sge0fsum  46358  sge0sup  46362  sge0gerp  46366  sge0pnffigt  46367  sge0lefi  46369  sge0xaddlem1  46404  sge0xaddlem2  46405  meadjiunlem  46436  sinnpoly  46865