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Theorem rnmptssdf 42753
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
rnmptssdf.1 𝑥𝜑
rnmptssdf.2 𝑥𝐶
rnmptssdf.3 𝐹 = (𝑥𝐴𝐵)
rnmptssdf.4 ((𝜑𝑥𝐴) → 𝐵𝐶)
Assertion
Ref Expression
rnmptssdf (𝜑 → ran 𝐹𝐶)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐶(𝑥)   𝐹(𝑥)

Proof of Theorem rnmptssdf
StepHypRef Expression
1 rnmptssdf.1 . . 3 𝑥𝜑
2 rnmptssdf.4 . . 3 ((𝜑𝑥𝐴) → 𝐵𝐶)
31, 2ralrimia 3428 . 2 (𝜑 → ∀𝑥𝐴 𝐵𝐶)
4 rnmptssdf.2 . . 3 𝑥𝐶
5 rnmptssdf.3 . . 3 𝐹 = (𝑥𝐴𝐵)
64, 5rnmptssf 42746 . 2 (∀𝑥𝐴 𝐵𝐶 → ran 𝐹𝐶)
73, 6syl 17 1 (𝜑 → ran 𝐹𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1541  wnf 1789  wcel 2109  wnfc 2888  wral 3065  wss 3891  cmpt 5161  ran crn 5589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pr 5355
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-sbc 3720  df-csb 3837  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-fun 6432  df-fn 6433  df-f 6434
This theorem is referenced by:  rnmptss2  42756  supminfrnmpt  42939  supminfxrrnmpt  42965
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