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Theorem rnmptssd 45138
Description: The range of a function given by the maps-to notation as a subset. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
Hypotheses
Ref Expression
rnmptssd.1 𝑥𝜑
rnmptssd.2 𝐹 = (𝑥𝐴𝐵)
rnmptssd.3 ((𝜑𝑥𝐴) → 𝐵𝐶)
Assertion
Ref Expression
rnmptssd (𝜑 → ran 𝐹𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem rnmptssd
StepHypRef Expression
1 rnmptssd.1 . . 3 𝑥𝜑
2 rnmptssd.3 . . 3 ((𝜑𝑥𝐴) → 𝐵𝐶)
31, 2ralrimia 3255 . 2 (𝜑 → ∀𝑥𝐴 𝐵𝐶)
4 rnmptssd.2 . . 3 𝐹 = (𝑥𝐴𝐵)
54rnmptss 7142 . 2 (∀𝑥𝐴 𝐵𝐶 → ran 𝐹𝐶)
63, 5syl 17 1 (𝜑 → ran 𝐹𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1536  wnf 1779  wcel 2105  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-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:  infnsuprnmpt  45194  suprclrnmpt  45195  suprubrnmpt2  45196  suprubrnmpt  45197  fisupclrnmpt  45347  supxrleubrnmpt  45355  infxrlbrnmpt2  45359  supxrrernmpt  45370  suprleubrnmpt  45371  infrnmptle  45372  infxrunb3rnmpt  45377  supxrre3rnmpt  45378  supminfrnmpt  45394  infxrrnmptcl  45396  infxrgelbrnmpt  45403  infrpgernmpt  45414  supminfxrrnmpt  45420  liminfcl  45718  fourierdlem31  46093  fourierdlem53  46114  sge0xaddlem2  46389  sge0reuz  46402  sge0reuzb  46403  meadjiun  46421  hoidmvlelem2  46551  iunhoiioolem  46630  vonioolem1  46635  smflimsuplem4  46778
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