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Theorem rnmptssff 43402
Description: The range of a function given by the maps-to notation as a subset. (Contributed by Glauco Siliprandi, 24-Jan-2025.)
Hypotheses
Ref Expression
rnmptssff.1 𝑥𝐴
rnmptssff.2 𝑥𝐶
rnmptssff.3 𝐹 = (𝑥𝐴𝐵)
Assertion
Ref Expression
rnmptssff (∀𝑥𝐴 𝐵𝐶 → ran 𝐹𝐶)

Proof of Theorem rnmptssff
StepHypRef Expression
1 rnmptssff.1 . . 3 𝑥𝐴
2 rnmptssff.2 . . 3 𝑥𝐶
3 rnmptssff.3 . . 3 𝐹 = (𝑥𝐴𝐵)
41, 2, 3fmptff 43397 . 2 (∀𝑥𝐴 𝐵𝐶𝐹:𝐴𝐶)
5 frn 6672 . 2 (𝐹:𝐴𝐶 → ran 𝐹𝐶)
64, 5sylbi 216 1 (∀𝑥𝐴 𝐵𝐶 → ran 𝐹𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  wcel 2106  wnfc 2885  wral 3062  wss 3908  cmpt 5186  ran crn 5632  wf 6489
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-sbc 3738  df-csb 3854  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-fun 6495  df-fn 6496  df-f 6497
This theorem is referenced by:  rnmptssdff  43403
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