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Theorem rnmptss2 42324
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 5125 . . 3 𝑥(𝑥𝐵𝐶)
32nfrn 5789 . 2 𝑥ran (𝑥𝐵𝐶)
4 eqid 2738 . 2 (𝑥𝐴𝐶) = (𝑥𝐴𝐶)
5 eqid 2738 . . 3 (𝑥𝐵𝐶) = (𝑥𝐵𝐶)
6 rnmptss2.3 . . . 4 (𝜑𝐴𝐵)
76sselda 3875 . . 3 ((𝜑𝑥𝐴) → 𝑥𝐵)
8 rnmptss2.4 . . 3 ((𝜑𝑥𝐴) → 𝐶𝑉)
95, 7, 8elrnmpt1d 42295 . 2 ((𝜑𝑥𝐴) → 𝐶 ∈ ran (𝑥𝐵𝐶))
101, 3, 4, 9rnmptssdf 42321 1 (𝜑 → ran (𝑥𝐴𝐶) ⊆ ran (𝑥𝐵𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399  wnf 1790  wcel 2113  wss 3841  cmpt 5107  ran crn 5520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-sep 5164  ax-nul 5171  ax-pr 5293
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-sn 4514  df-pr 4516  df-op 4520  df-br 5028  df-opab 5090  df-mpt 5108  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-fun 6335  df-fn 6336  df-f 6337
This theorem is referenced by:  smflimsuplem4  43879
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