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  20723  smadiadetlem3lem2  22587  tgiun  22899  ustuqtop0  24161  metustss  24472  efabl  26492  efsubm  26493  fnpreimac  32645  prodindf  32836  swrdrn2  32926  gsummpt2co  33031  psgnfzto1stlem  33072  elrgspnsubrunlem1  33214  nsgmgc  33376  nsgqusf1olem1  33377  algextdeglem2  33701  algextdeglem4  33703  locfinreflem  33823  rspectopn  33850  zarcls  33857  zartopn  33858  gsumesum  34042  esumlub  34043  esumgect  34073  esum2d  34076  ldgenpisyslem1  34146  sxbrsigalem0  34255  omscl  34279  omsmon  34282  carsgclctunlem2  34303  carsgclctunlem3  34304  pmeasadd  34309  hgt750lemb  34640  mnurndlem2  44264  suprnmpt  45161  rnmptssrn  45169  wessf1ornlem  45172  rnmptssd  45183  rnmptssbi  45247  liminflelimsuplem  45766  fourierdlem53  46150  fourierdlem111  46208  ioorrnopnlem  46295  salexct3  46333  salgensscntex  46335  sge0rnre  46355  sge0tsms  46371  sge0cl  46372  sge0fsum  46378  sge0sup  46382  sge0gerp  46386  sge0pnffigt  46387  sge0lefi  46389  sge0xaddlem1  46424  sge0xaddlem2  46425  meadjiunlem  46456  sinnpoly  46885