Theorem rnmptss 7095

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 7082	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ↔ 𝐹:𝐴⟶𝐶)
3		frn 6695	. 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 3914   ↦ cmpt 5188  ran crn 5639  ⟶wf 6507

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 5251  ax-nul 5261  ax-pr 5387

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 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-fun 6513  df-fn 6514  df-f 6515

This theorem is referenced by:  mptexw  7931  iunon  8308  iinon  8309  gruiun  10752  subdrgint  20712  smadiadetlem3lem2  22554  tgiun  22866  ustuqtop0  24128  metustss  24439  efabl  26459  efsubm  26460  fnpreimac  32595  prodindf  32786  swrdrn2  32876  gsummpt2co  32988  psgnfzto1stlem  33057  elrgspnsubrunlem1  33198  nsgmgc  33383  nsgqusf1olem1  33384  algextdeglem2  33708  algextdeglem4  33710  locfinreflem  33830  rspectopn  33857  zarcls  33864  zartopn  33865  gsumesum  34049  esumlub  34050  esumgect  34080  esum2d  34083  ldgenpisyslem1  34153  sxbrsigalem0  34262  omscl  34286  omsmon  34289  carsgclctunlem2  34310  carsgclctunlem3  34311  pmeasadd  34316  hgt750lemb  34647  mnurndlem2  44271  suprnmpt  45168  rnmptssrn  45176  wessf1ornlem  45179  rnmptssd  45190  rnmptssbi  45254  liminflelimsuplem  45773  fourierdlem53  46157  fourierdlem111  46215  ioorrnopnlem  46302  salexct3  46340  salgensscntex  46342  sge0rnre  46362  sge0tsms  46378  sge0cl  46379  sge0fsum  46385  sge0sup  46389  sge0gerp  46393  sge0pnffigt  46394  sge0lefi  46396  sge0xaddlem1  46431  sge0xaddlem2  46432  meadjiunlem  46463  sinnpoly  46892