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Theorem rnmptssdff 45255
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
rnmptssdff.1 𝑥𝜑
rnmptssdff.2 𝑥𝐴
rnmptssdff.3 𝑥𝐶
rnmptssdff.4 𝐹 = (𝑥𝐴𝐵)
rnmptssdff.5 ((𝜑𝑥𝐴) → 𝐵𝐶)
Assertion
Ref Expression
rnmptssdff (𝜑 → ran 𝐹𝐶)

Proof of Theorem rnmptssdff
StepHypRef Expression
1 rnmptssdff.1 . . 3 𝑥𝜑
2 rnmptssdff.5 . . 3 ((𝜑𝑥𝐴) → 𝐵𝐶)
31, 2ralrimia 3257 . 2 (𝜑 → ∀𝑥𝐴 𝐵𝐶)
4 rnmptssdff.2 . . 3 𝑥𝐴
5 rnmptssdff.3 . . 3 𝑥𝐶
6 rnmptssdff.4 . . 3 𝐹 = (𝑥𝐴𝐵)
74, 5, 6rnmptssff 45254 . 2 (∀𝑥𝐴 𝐵𝐶 → ran 𝐹𝐶)
83, 7syl 17 1 (𝜑 → ran 𝐹𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wnf 1783  wcel 2108  wnfc 2889  wral 3060  wss 3950  cmpt 5223  ran crn 5684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5294  ax-nul 5304  ax-pr 5430
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5142  df-opab 5204  df-mpt 5224  df-id 5576  df-xp 5689  df-rel 5690  df-cnv 5691  df-co 5692  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-fun 6561  df-fn 6562  df-f 6563
This theorem is referenced by:  saliunclf  46310
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