Theorem rnmptss 7075

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 7062	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ↔ 𝐹:𝐴⟶𝐶)
3		frn 6675	. 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 1542   ∈ wcel 2114  ∀wral 3051   ⊆ wss 3889   ↦ cmpt 5166  ran crn 5632  ⟶wf 6494

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-mpt 5167  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 6500  df-fn 6501  df-f 6502

This theorem is referenced by:  rnmptssd  7076  mptexw  7906  iunon  8279  iinon  8280  gruiun  10722  subdrgint  20780  smadiadetlem3lem2  22632  tgiun  22944  ustuqtop0  24205  metustss  24516  efabl  26514  efsubm  26515  fnpreimac  32743  prodindf  32922  swrdrn2  33014  gsummpt2co  33109  psgnfzto1stlem  33161  elrgspnsubrunlem1  33308  nsgmgc  33472  nsgqusf1olem1  33473  algextdeglem2  33862  algextdeglem4  33864  locfinreflem  33984  rspectopn  34011  zarcls  34018  zartopn  34019  gsumesum  34203  esumlub  34204  esumgect  34234  esum2d  34237  ldgenpisyslem1  34307  sxbrsigalem0  34415  omscl  34439  omsmon  34442  carsgclctunlem2  34463  carsgclctunlem3  34464  pmeasadd  34469  hgt750lemb  34800  mnurndlem2  44709  suprnmpt  45604  rnmptssrn  45612  wessf1ornlem  45615  rnmptssbi  45689  liminflelimsuplem  46203  fourierdlem53  46587  fourierdlem111  46645  ioorrnopnlem  46732  salexct3  46770  salgensscntex  46772  sge0rnre  46792  sge0tsms  46808  sge0cl  46809  sge0fsum  46815  sge0sup  46819  sge0gerp  46823  sge0pnffigt  46824  sge0lefi  46826  sge0xaddlem1  46861  sge0xaddlem2  46862  meadjiunlem  46893  sinnpoly  47339