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Theorem mptima2 41821
Description: Image of a function in maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypothesis
Ref Expression
mptima2.1 (𝜑𝐶𝐴)
Assertion
Ref Expression
mptima2 (𝜑 → ((𝑥𝐴𝐵) “ 𝐶) = ran (𝑥𝐶𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem mptima2
StepHypRef Expression
1 mptima 5919 . 2 ((𝑥𝐴𝐵) “ 𝐶) = ran (𝑥 ∈ (𝐴𝐶) ↦ 𝐵)
2 mptima2.1 . . . . 5 (𝜑𝐶𝐴)
3 sseqin2 4166 . . . . 5 (𝐶𝐴 ↔ (𝐴𝐶) = 𝐶)
42, 3sylib 221 . . . 4 (𝜑 → (𝐴𝐶) = 𝐶)
54mpteq1d 5131 . . 3 (𝜑 → (𝑥 ∈ (𝐴𝐶) ↦ 𝐵) = (𝑥𝐶𝐵))
65rneqd 5785 . 2 (𝜑 → ran (𝑥 ∈ (𝐴𝐶) ↦ 𝐵) = ran (𝑥𝐶𝐵))
71, 6syl5eq 2869 1 (𝜑 → ((𝑥𝐴𝐵) “ 𝐶) = ran (𝑥𝐶𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1538  cin 3907  wss 3908  cmpt 5122  ran crn 5533  cima 5535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rab 3139  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-br 5043  df-opab 5105  df-mpt 5123  df-xp 5538  df-rel 5539  df-cnv 5540  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545
This theorem is referenced by:  limsupresico  42281  limsupvaluz  42289  liminfresico  42352
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