Theorem rnmptss 7122

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 7110	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ↔ 𝐹:𝐴⟶𝐶)
3		frn 6725	. 2 ⊢ (𝐹:𝐴⟶𝐶 → ran 𝐹 ⊆ 𝐶)
4	2, 3	sylbi 216	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ran 𝐹 ⊆ 𝐶)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2107  ∀wral 3062   ⊆ wss 3949   ↦ cmpt 5232  ran crn 5678  ⟶wf 6540

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-fun 6546  df-fn 6547  df-f 6548

This theorem is referenced by:  mptexw  7939  iunon  8339  iinon  8340  gruiun  10794  subdrgint  20419  smadiadetlem3lem2  22169  tgiun  22482  ustuqtop0  23745  metustss  24060  efabl  26059  efsubm  26060  fnpreimac  31927  swrdrn2  32149  gsummpt2co  32231  psgnfzto1stlem  32290  nsgmgc  32554  nsgqusf1olem1  32555  algextdeglem1  32803  locfinreflem  32851  rspectopn  32878  zarcls  32885  zartopn  32886  prodindf  33052  gsumesum  33088  esumlub  33089  esumgect  33119  esum2d  33122  ldgenpisyslem1  33192  sxbrsigalem0  33301  omscl  33325  omsmon  33328  carsgclctunlem2  33349  carsgclctunlem3  33350  pmeasadd  33355  hgt750lemb  33699  mnurndlem2  43089  suprnmpt  43918  rnmptssrn  43927  wessf1ornlem  43930  rnmptssd  43943  rnmptssbi  44013  liminflelimsuplem  44539  fourierdlem53  44923  fourierdlem111  44981  ioorrnopnlem  45068  salexct3  45106  salgensscntex  45108  sge0rnre  45128  sge0tsms  45144  sge0cl  45145  sge0fsum  45151  sge0sup  45155  sge0gerp  45159  sge0pnffigt  45160  sge0lefi  45162  sge0xaddlem1  45197  sge0xaddlem2  45198  meadjiunlem  45229