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Theorem rnmptssd 42779
Description: The range of an operation 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 3238 . 2 (𝜑 → ∀𝑥𝐴 𝐵𝐶)
4 rnmptssd.2 . . 3 𝐹 = (𝑥𝐴𝐵)
54rnmptss 7028 . 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 1539  wnf 1783  wcel 2104  wral 3062  wss 3892  cmpt 5164  ran crn 5601
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pr 5361
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ral 3063  df-rex 3072  df-rab 3287  df-v 3439  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-sn 4566  df-pr 4568  df-op 4572  df-br 5082  df-opab 5144  df-mpt 5165  df-id 5500  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-rn 5611  df-res 5612  df-ima 5613  df-fun 6460  df-fn 6461  df-f 6462
This theorem is referenced by:  infnsuprnmpt  42841  suprclrnmpt  42842  suprubrnmpt2  42843  suprubrnmpt  42844  fisupclrnmpt  42986  supxrleubrnmpt  42994  infxrlbrnmpt2  42998  supxrrernmpt  43009  suprleubrnmpt  43010  infrnmptle  43011  infxrunb3rnmpt  43016  supxrre3rnmpt  43017  supminfrnmpt  43033  infxrrnmptcl  43035  infxrgelbrnmpt  43042  infrpgernmpt  43053  supminfxrrnmpt  43059  liminfcl  43353  fourierdlem31  43728  fourierdlem53  43749  sge0xaddlem2  44022  sge0reuz  44035  sge0reuzb  44036  meadjiun  44054  hoidmvlelem2  44184  iunhoiioolem  44263  vonioolem1  44268  smflimsuplem4  44410
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