Theorem rnmptss 6896

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 6884	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ↔ 𝐹:𝐴⟶𝐶)
3		frn 6511	. 2 ⊢ (𝐹:𝐴⟶𝐶 → ran 𝐹 ⊆ 𝐶)
4	2, 3	sylbi 220	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ran 𝐹 ⊆ 𝐶)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  ∀wral 3053   ⊆ wss 3843   ↦ cmpt 5110  ran crn 5526  ⟶wf 6335

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-fun 6341  df-fn 6342  df-f 6343

This theorem is referenced by:  mptexw  7679  iunon  8005  iinon  8006  gruiun  10299  subdrgint  19701  smadiadetlem3lem2  21418  tgiun  21730  ustuqtop0  22992  metustss  23304  efabl  25294  efsubm  25295  fnpreimac  30583  swrdrn2  30801  gsummpt2co  30885  psgnfzto1stlem  30944  nsgmgc  31169  nsgqusf1olem1  31170  locfinreflem  31362  rspectopn  31389  zarcls  31396  zartopn  31397  prodindf  31561  gsumesum  31597  esumlub  31598  esumgect  31628  esum2d  31631  ldgenpisyslem1  31701  sxbrsigalem0  31808  omscl  31832  omsmon  31835  carsgclctunlem2  31856  carsgclctunlem3  31857  pmeasadd  31862  hgt750lemb  32206  mnurndlem2  41442  suprnmpt  42248  rnmptssrn  42257  wessf1ornlem  42260  rnmptssd  42273  rnmptssbi  42344  liminflelimsuplem  42858  fourierdlem53  43242  fourierdlem111  43300  ioorrnopnlem  43387  saliuncl  43405  salexct3  43423  salgensscntex  43425  sge0rnre  43444  sge0tsms  43460  sge0cl  43461  sge0fsum  43467  sge0sup  43471  sge0gerp  43475  sge0pnffigt  43476  sge0lefi  43478  sge0xaddlem1  43513  sge0xaddlem2  43514  meadjiunlem  43545