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Theorem rnmptssff 44714
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 44709 . 2 (∀𝑥𝐴 𝐵𝐶𝐹:𝐴𝐶)
5 frn 6724 . 2 (𝐹:𝐴𝐶 → ran 𝐹𝐶)
64, 5sylbi 216 1 (∀𝑥𝐴 𝐵𝐶 → ran 𝐹𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  wnfc 2875  wral 3051  wss 3939  cmpt 5226  ran crn 5673  wf 6539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-fun 6545  df-fn 6546  df-f 6547
This theorem is referenced by:  rnmptssdff  44715
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