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Theorem rnmptssdf 45198
Description: The range of a function 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 3255 . 2 (𝜑 → ∀𝑥𝐴 𝐵𝐶)
4 rnmptssdf.2 . . 3 𝑥𝐶
5 rnmptssdf.3 . . 3 𝐹 = (𝑥𝐴𝐵)
64, 5rnmptssf 45191 . 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 1536  wnf 1779  wcel 2105  wnfc 2887  wral 3058  wss 3962  cmpt 5230  ran crn 5689
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-fun 6564  df-fn 6565  df-f 6566
This theorem is referenced by:  rnmptss2  45201  supminfrnmpt  45394  supminfxrrnmpt  45420
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