Theorem rnmptss 6885

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 6873	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ↔ 𝐹:𝐴⟶𝐶)
3		frn 6519	. 2 ⊢ (𝐹:𝐴⟶𝐶 → ran 𝐹 ⊆ 𝐶)
4	2, 3	sylbi 219	1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ran 𝐹 ⊆ 𝐶)

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2110  ∀wral 3138   ⊆ wss 3935   ↦ cmpt 5145  ran crn 5555  ⟶wf 6350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793  ax-sep 5202  ax-nul 5209  ax-pr 5329

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4838  df-br 5066  df-opab 5128  df-mpt 5146  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-fv 6362

This theorem is referenced by:  mptexw  7653  iunon  7975  iinon  7976  gruiun  10220  subdrgint  19581  smadiadetlem3lem2  21275  tgiun  21586  ustuqtop0  22848  metustss  23160  efabl  25133  efsubm  25134  fnpreimac  30415  swrdrn2  30628  gsummpt2co  30686  psgnfzto1stlem  30742  locfinreflem  31104  prodindf  31282  gsumesum  31318  esumlub  31319  esumgect  31349  esum2d  31352  ldgenpisyslem1  31422  sxbrsigalem0  31529  omscl  31553  omsmon  31556  carsgclctunlem2  31577  carsgclctunlem3  31578  pmeasadd  31583  hgt750lemb  31927  mnurndlem2  40616  suprnmpt  41428  rnmptssrn  41440  wessf1ornlem  41443  rnmptssd  41456  rnmptssbi  41532  liminflelimsuplem  42054  fourierdlem31  42422  fourierdlem53  42443  fourierdlem111  42501  ioorrnopnlem  42588  saliuncl  42606  salexct3  42624  salgensscntex  42626  sge0rnre  42645  sge0tsms  42661  sge0cl  42662  sge0fsum  42668  sge0sup  42672  sge0gerp  42676  sge0pnffigt  42677  sge0lefi  42679  sge0xaddlem1  42714  sge0xaddlem2  42715  meadjiunlem  42746  meadjiun  42747