Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rnmptssdf Structured version   Visualization version   GIF version

Theorem rnmptssdf 44693
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 3246 . 2 (𝜑 → ∀𝑥𝐴 𝐵𝐶)
4 rnmptssdf.2 . . 3 𝑥𝐶
5 rnmptssdf.3 . . 3 𝐹 = (𝑥𝐴𝐵)
64, 5rnmptssf 44686 . 2 (∀𝑥𝐴 𝐵𝐶 → ran 𝐹𝐶)
73, 6syl 17 1 (𝜑 → ran 𝐹𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394   = wceq 1533  wnf 1777  wcel 2098  wnfc 2875  wral 3051  wss 3939  cmpt 5226  ran crn 5673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-sbc 3769  df-csb 3885  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-fun 6545  df-fn 6546  df-f 6547
This theorem is referenced by:  rnmptss2  44696  supminfrnmpt  44890  supminfxrrnmpt  44916
  Copyright terms: Public domain W3C validator