Theorem rnmptss 7113

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 7100	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ↔ 𝐹:𝐴⟶𝐶)
3		frn 6713	. 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 2108  ∀wral 3051   ⊆ wss 3926   ↦ cmpt 5201  ran crn 5655  ⟶wf 6527

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

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 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-fun 6533  df-fn 6534  df-f 6535

This theorem is referenced by:  mptexw  7951  iunon  8353  iinon  8354  gruiun  10813  subdrgint  20763  smadiadetlem3lem2  22605  tgiun  22917  ustuqtop0  24179  metustss  24490  efabl  26511  efsubm  26512  fnpreimac  32649  prodindf  32840  swrdrn2  32930  gsummpt2co  33042  psgnfzto1stlem  33111  elrgspnsubrunlem1  33242  nsgmgc  33427  nsgqusf1olem1  33428  algextdeglem2  33752  algextdeglem4  33754  locfinreflem  33871  rspectopn  33898  zarcls  33905  zartopn  33906  gsumesum  34090  esumlub  34091  esumgect  34121  esum2d  34124  ldgenpisyslem1  34194  sxbrsigalem0  34303  omscl  34327  omsmon  34330  carsgclctunlem2  34351  carsgclctunlem3  34352  pmeasadd  34357  hgt750lemb  34688  mnurndlem2  44306  suprnmpt  45198  rnmptssrn  45206  wessf1ornlem  45209  rnmptssd  45220  rnmptssbi  45284  liminflelimsuplem  45804  fourierdlem53  46188  fourierdlem111  46246  ioorrnopnlem  46333  salexct3  46371  salgensscntex  46373  sge0rnre  46393  sge0tsms  46409  sge0cl  46410  sge0fsum  46416  sge0sup  46420  sge0gerp  46424  sge0pnffigt  46425  sge0lefi  46427  sge0xaddlem1  46462  sge0xaddlem2  46463  meadjiunlem  46494