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Theorem rnmptssf 41013
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 41003 . 2 (∀𝑥𝐴 𝐵𝐶𝐹:𝐴𝐶)
4 frn 6380 . 2 (𝐹:𝐴𝐶 → ran 𝐹𝐶)
53, 4sylbi 218 1 (∀𝑥𝐴 𝐵𝐶 → ran 𝐹𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1520  wcel 2079  wnfc 2931  wral 3103  wss 3854  cmpt 5035  ran crn 5436  wf 6213
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1775  ax-4 1789  ax-5 1886  ax-6 1945  ax-7 1990  ax-8 2081  ax-9 2089  ax-10 2110  ax-11 2124  ax-12 2139  ax-13 2342  ax-ext 2767  ax-sep 5088  ax-nul 5095  ax-pr 5214
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1080  df-tru 1523  df-ex 1760  df-nf 1764  df-sb 2041  df-mo 2574  df-eu 2610  df-clab 2774  df-cleq 2786  df-clel 2861  df-nfc 2933  df-ne 2983  df-ral 3108  df-rex 3109  df-rab 3112  df-v 3434  df-sbc 3702  df-csb 3807  df-dif 3857  df-un 3859  df-in 3861  df-ss 3869  df-nul 4207  df-if 4376  df-sn 4467  df-pr 4469  df-op 4473  df-uni 4740  df-br 4957  df-opab 5019  df-mpt 5036  df-id 5340  df-xp 5441  df-rel 5442  df-cnv 5443  df-co 5444  df-dm 5445  df-rn 5446  df-res 5447  df-ima 5448  df-iota 6181  df-fun 6219  df-fn 6220  df-f 6221  df-fv 6225
This theorem is referenced by:  rnmptssdf  41020
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