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Theorem rnmptss2 42803
Description: The range of an operation 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 5182 . . 3 𝑥(𝑥𝐵𝐶)
32nfrn 5861 . 2 𝑥ran (𝑥𝐵𝐶)
4 eqid 2738 . 2 (𝑥𝐴𝐶) = (𝑥𝐴𝐶)
5 eqid 2738 . . 3 (𝑥𝐵𝐶) = (𝑥𝐵𝐶)
6 rnmptss2.3 . . . 4 (𝜑𝐴𝐵)
76sselda 3921 . . 3 ((𝜑𝑥𝐴) → 𝑥𝐵)
8 rnmptss2.4 . . 3 ((𝜑𝑥𝐴) → 𝐶𝑉)
95, 7, 8elrnmpt1d 42773 . 2 ((𝜑𝑥𝐴) → 𝐶 ∈ ran (𝑥𝐵𝐶))
101, 3, 4, 9rnmptssdf 42800 1 (𝜑 → ran (𝑥𝐴𝐶) ⊆ ran (𝑥𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  wnf 1786  wcel 2106  wss 3887  cmpt 5157  ran crn 5590
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-fun 6435  df-fn 6436  df-f 6437
This theorem is referenced by:  smflimsuplem4  44356
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