Theorem rnmptss 6299

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 6289	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ↔ 𝐹:𝐴⟶𝐶)
3		frn 5966	. 2 ⊢ (𝐹:𝐴⟶𝐶 → ran 𝐹 ⊆ 𝐶)
4	2, 3	sylbi 206	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ran 𝐹 ⊆ 𝐶)

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  ∀wral 2896   ⊆ wss 3540   ↦ cmpt 4643  ran crn 5039  ⟶wf 5800

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812

This theorem is referenced by:  iunon  7323  iinon  7324  gruiun  9500  smadiadetlem3lem2  20292  tgiun  20594  ustuqtop0  21854  metustss  22166  efabl  24100  efsubm  24101  gsummpt2co  29111  psgnfzto1stlem  29181  locfinreflem  29235  gsumesum  29448  esumlub  29449  esumgect  29479  esum2d  29482  ldgenpisyslem1  29553  sxbrsigalem0  29660  omscl  29684  omsmon  29687  carsgclctunlem2  29708  carsgclctunlem3  29709  pmeasadd  29714  suprnmpt  38350  rnmptssrn  38363  wessf1ornlem  38366  rnmptssd  38380  fourierdlem31  39031  fourierdlem53  39052  fourierdlem111  39110  ioorrnopnlem  39200  saliuncl  39218  salexct3  39236  salgensscntex  39238  sge0rnre  39257  sge0tsms  39273  sge0cl  39274  sge0fsum  39280  sge0sup  39284  sge0gerp  39288  sge0pnffigt  39289  sge0lefi  39291  sge0xaddlem1  39326  sge0xaddlem2  39327  meadjiunlem  39358  meadjiun  39359