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Theorem rnmptss 5621
 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 = (𝑥𝐴𝐵)
Assertion
Ref Expression
rnmptss (∀𝑥𝐴 𝐵𝐶 → ran 𝐹𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶
Allowed substitution hints:   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem rnmptss
StepHypRef Expression
1 rnmptss.1 . . 3 𝐹 = (𝑥𝐴𝐵)
21fmpt 5610 . 2 (∀𝑥𝐴 𝐵𝐶𝐹:𝐴𝐶)
3 frn 5321 . 2 (𝐹:𝐴𝐶 → ran 𝐹𝐶)
42, 3sylbi 120 1 (∀𝑥𝐴 𝐵𝐶 → ran 𝐹𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1332   ∈ wcel 2125  ∀wral 2432   ⊆ wss 3098   ↦ cmpt 4021  ran crn 4580  ⟶wf 5159 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-pow 4130  ax-pr 4164 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ral 2437  df-rex 2438  df-rab 2441  df-v 2711  df-sbc 2934  df-un 3102  df-in 3104  df-ss 3111  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-uni 3769  df-br 3962  df-opab 4022  df-mpt 4023  df-id 4248  df-xp 4585  df-rel 4586  df-cnv 4587  df-co 4588  df-dm 4589  df-rn 4590  df-res 4591  df-ima 4592  df-iota 5128  df-fun 5165  df-fn 5166  df-f 5167  df-fv 5171 This theorem is referenced by:  tgiun  12420  dvrecap  13024
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