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Theorem rnmptssd 43895
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 3256 . 2 (𝜑 → ∀𝑥𝐴 𝐵𝐶)
4 rnmptssd.2 . . 3 𝐹 = (𝑥𝐴𝐵)
54rnmptss 7122 . 2 (∀𝑥𝐴 𝐵𝐶 → ran 𝐹𝐶)
63, 5syl 17 1 (𝜑 → ran 𝐹𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wnf 1786  wcel 2107  wral 3062  wss 3949  cmpt 5232  ran crn 5678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-fun 6546  df-fn 6547  df-f 6548
This theorem is referenced by:  infnsuprnmpt  43954  suprclrnmpt  43955  suprubrnmpt2  43956  suprubrnmpt  43957  fisupclrnmpt  44108  supxrleubrnmpt  44116  infxrlbrnmpt2  44120  supxrrernmpt  44131  suprleubrnmpt  44132  infrnmptle  44133  infxrunb3rnmpt  44138  supxrre3rnmpt  44139  supminfrnmpt  44155  infxrrnmptcl  44157  infxrgelbrnmpt  44164  infrpgernmpt  44175  supminfxrrnmpt  44181  liminfcl  44479  fourierdlem31  44854  fourierdlem53  44875  sge0xaddlem2  45150  sge0reuz  45163  sge0reuzb  45164  meadjiun  45182  hoidmvlelem2  45312  iunhoiioolem  45391  vonioolem1  45396  smflimsuplem4  45539
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