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Theorem rnmptssf 41883
 Description: The range of an operation given by the maps-to notation as a subset. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
rnmptssf.1 𝑥𝐶
rnmptssf.2 𝐹 = (𝑥𝐴𝐵)
Assertion
Ref Expression
rnmptssf (∀𝑥𝐴 𝐵𝐶 → ran 𝐹𝐶)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)   𝐹(𝑥)

Proof of Theorem rnmptssf
StepHypRef Expression
1 rnmptssf.1 . . 3 𝑥𝐶
2 rnmptssf.2 . . 3 𝐹 = (𝑥𝐴𝐵)
31, 2fmptf 41873 . 2 (∀𝑥𝐴 𝐵𝐶𝐹:𝐴𝐶)
4 frn 6493 . 2 (𝐹:𝐴𝐶 → ran 𝐹𝐶)
53, 4sylbi 220 1 (∀𝑥𝐴 𝐵𝐶 → ran 𝐹𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  Ⅎwnfc 2936  ∀wral 3106   ⊆ wss 3881   ↦ cmpt 5110  ran crn 5520  ⟶wf 6320 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332 This theorem is referenced by:  rnmptssdf  41890
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