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Theorem imaindm 33017
Description: The image is unaffected by intersection with the domain. (Contributed by Scott Fenton, 17-Dec-2021.)
Assertion
Ref Expression
imaindm (𝑅𝐴) = (𝑅 “ (𝐴 ∩ dom 𝑅))

Proof of Theorem imaindm
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3498 . . . . . . 7 𝑦 ∈ V
2 vex 3498 . . . . . . 7 𝑥 ∈ V
31, 2breldm 5772 . . . . . 6 (𝑦𝑅𝑥𝑦 ∈ dom 𝑅)
43pm4.71ri 563 . . . . 5 (𝑦𝑅𝑥 ↔ (𝑦 ∈ dom 𝑅𝑦𝑅𝑥))
54rexbii 3247 . . . 4 (∃𝑦𝐴 𝑦𝑅𝑥 ↔ ∃𝑦𝐴 (𝑦 ∈ dom 𝑅𝑦𝑅𝑥))
6 rexin 4216 . . . 4 (∃𝑦 ∈ (𝐴 ∩ dom 𝑅)𝑦𝑅𝑥 ↔ ∃𝑦𝐴 (𝑦 ∈ dom 𝑅𝑦𝑅𝑥))
75, 6bitr4i 280 . . 3 (∃𝑦𝐴 𝑦𝑅𝑥 ↔ ∃𝑦 ∈ (𝐴 ∩ dom 𝑅)𝑦𝑅𝑥)
82elima 5929 . . 3 (𝑥 ∈ (𝑅𝐴) ↔ ∃𝑦𝐴 𝑦𝑅𝑥)
92elima 5929 . . 3 (𝑥 ∈ (𝑅 “ (𝐴 ∩ dom 𝑅)) ↔ ∃𝑦 ∈ (𝐴 ∩ dom 𝑅)𝑦𝑅𝑥)
107, 8, 93bitr4i 305 . 2 (𝑥 ∈ (𝑅𝐴) ↔ 𝑥 ∈ (𝑅 “ (𝐴 ∩ dom 𝑅)))
1110eqriv 2818 1 (𝑅𝐴) = (𝑅 “ (𝐴 ∩ dom 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wa 398   = wceq 1533  wcel 2110  wrex 3139  cin 3935   class class class wbr 5059  dom cdm 5550  cima 5553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pr 5322
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-br 5060  df-opab 5122  df-xp 5556  df-cnv 5558  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563
This theorem is referenced by:  madeval2  33285
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