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Theorem mptima2 39771
Description: Image of a function in map-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 39751 . . 3 ((𝑥𝐴𝐵) “ 𝐶) = ran (𝑥 ∈ (𝐴𝐶) ↦ 𝐵)
21a1i 11 . 2 (𝜑 → ((𝑥𝐴𝐵) “ 𝐶) = ran (𝑥 ∈ (𝐴𝐶) ↦ 𝐵))
3 mptima2.1 . . . . 5 (𝜑𝐶𝐴)
4 sseqin2 3850 . . . . . 6 (𝐶𝐴 ↔ (𝐴𝐶) = 𝐶)
54biimpi 206 . . . . 5 (𝐶𝐴 → (𝐴𝐶) = 𝐶)
63, 5syl 17 . . . 4 (𝜑 → (𝐴𝐶) = 𝐶)
76mpteq1d 4771 . . 3 (𝜑 → (𝑥 ∈ (𝐴𝐶) ↦ 𝐵) = (𝑥𝐶𝐵))
87rneqd 5385 . 2 (𝜑 → ran (𝑥 ∈ (𝐴𝐶) ↦ 𝐵) = ran (𝑥𝐶𝐵))
92, 8eqtrd 2685 1 (𝜑 → ((𝑥𝐴𝐵) “ 𝐶) = ran (𝑥𝐶𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  cin 3606  wss 3607  cmpt 4762  ran crn 5144  cima 5146
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-mpt 4763  df-xp 5149  df-rel 5150  df-cnv 5151  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156
This theorem is referenced by:  limsupresico  40250  limsupvaluz  40258  liminfresico  40321
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