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Theorem mptima2 39956
 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 39936 . . 3 ((𝑥𝐴𝐵) “ 𝐶) = ran (𝑥 ∈ (𝐴𝐶) ↦ 𝐵)
21a1i 11 . 2 (𝜑 → ((𝑥𝐴𝐵) “ 𝐶) = ran (𝑥 ∈ (𝐴𝐶) ↦ 𝐵))
3 mptima2.1 . . . . 5 (𝜑𝐶𝐴)
4 sseqin2 3960 . . . . . 6 (𝐶𝐴 ↔ (𝐴𝐶) = 𝐶)
54biimpi 206 . . . . 5 (𝐶𝐴 → (𝐴𝐶) = 𝐶)
63, 5syl 17 . . . 4 (𝜑 → (𝐴𝐶) = 𝐶)
76mpteq1d 4890 . . 3 (𝜑 → (𝑥 ∈ (𝐴𝐶) ↦ 𝐵) = (𝑥𝐶𝐵))
87rneqd 5508 . 2 (𝜑 → ran (𝑥 ∈ (𝐴𝐶) ↦ 𝐵) = ran (𝑥𝐶𝐵))
92, 8eqtrd 2794 1 (𝜑 → ((𝑥𝐴𝐵) “ 𝐶) = ran (𝑥𝐶𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1632   ∩ cin 3714   ⊆ wss 3715   ↦ cmpt 4881  ran crn 5267   “ cima 5269 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-mpt 4882  df-xp 5272  df-rel 5273  df-cnv 5274  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279 This theorem is referenced by:  limsupresico  40435  limsupvaluz  40443  liminfresico  40506
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