Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rnmptss2 Structured version   Visualization version   GIF version

Theorem rnmptss2 45201
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
rnmptss2.1 𝑥𝜑
rnmptss2.3 (𝜑𝐴𝐵)
rnmptss2.4 ((𝜑𝑥𝐴) → 𝐶𝑉)
Assertion
Ref Expression
rnmptss2 (𝜑 → ran (𝑥𝐴𝐶) ⊆ ran (𝑥𝐵𝐶))
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐶(𝑥)   𝑉(𝑥)

Proof of Theorem rnmptss2
StepHypRef Expression
1 rnmptss2.1 . 2 𝑥𝜑
2 nfmpt1 5255 . . 3 𝑥(𝑥𝐵𝐶)
32nfrn 5965 . 2 𝑥ran (𝑥𝐵𝐶)
4 eqid 2734 . 2 (𝑥𝐴𝐶) = (𝑥𝐴𝐶)
5 eqid 2734 . . 3 (𝑥𝐵𝐶) = (𝑥𝐵𝐶)
6 rnmptss2.3 . . . 4 (𝜑𝐴𝐵)
76sselda 3994 . . 3 ((𝜑𝑥𝐴) → 𝑥𝐵)
8 rnmptss2.4 . . 3 ((𝜑𝑥𝐴) → 𝐶𝑉)
95, 7, 8elrnmpt1d 5977 . 2 ((𝜑𝑥𝐴) → 𝐶 ∈ ran (𝑥𝐵𝐶))
101, 3, 4, 9rnmptssdf 45198 1 (𝜑 → ran (𝑥𝐴𝐶) ⊆ ran (𝑥𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wnf 1779  wcel 2105  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:  smflimsuplem4  46778
  Copyright terms: Public domain W3C validator