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Theorem rnmptss2 45264
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 5250 . . 3 𝑥(𝑥𝐵𝐶)
32nfrn 5963 . 2 𝑥ran (𝑥𝐵𝐶)
4 eqid 2737 . 2 (𝑥𝐴𝐶) = (𝑥𝐴𝐶)
5 eqid 2737 . . 3 (𝑥𝐵𝐶) = (𝑥𝐵𝐶)
6 rnmptss2.3 . . . 4 (𝜑𝐴𝐵)
76sselda 3983 . . 3 ((𝜑𝑥𝐴) → 𝑥𝐵)
8 rnmptss2.4 . . 3 ((𝜑𝑥𝐴) → 𝐶𝑉)
95, 7, 8elrnmpt1d 5975 . 2 ((𝜑𝑥𝐴) → 𝐶 ∈ ran (𝑥𝐵𝐶))
101, 3, 4, 9rnmptssdf 45261 1 (𝜑 → ran (𝑥𝐴𝐶) ⊆ ran (𝑥𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wnf 1783  wcel 2108  wss 3951  cmpt 5225  ran crn 5686
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-fun 6563  df-fn 6564  df-f 6565
This theorem is referenced by:  smflimsuplem4  46838
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