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Theorem imaindm 6321
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 3482 . . . . . . 7 𝑦 ∈ V
2 vex 3482 . . . . . . 7 𝑥 ∈ V
31, 2breldm 5922 . . . . . 6 (𝑦𝑅𝑥𝑦 ∈ dom 𝑅)
43pm4.71ri 560 . . . . 5 (𝑦𝑅𝑥 ↔ (𝑦 ∈ dom 𝑅𝑦𝑅𝑥))
54rexbii 3092 . . . 4 (∃𝑦𝐴 𝑦𝑅𝑥 ↔ ∃𝑦𝐴 (𝑦 ∈ dom 𝑅𝑦𝑅𝑥))
6 rexin 4256 . . . 4 (∃𝑦 ∈ (𝐴 ∩ dom 𝑅)𝑦𝑅𝑥 ↔ ∃𝑦𝐴 (𝑦 ∈ dom 𝑅𝑦𝑅𝑥))
75, 6bitr4i 278 . . 3 (∃𝑦𝐴 𝑦𝑅𝑥 ↔ ∃𝑦 ∈ (𝐴 ∩ dom 𝑅)𝑦𝑅𝑥)
82elima 6085 . . 3 (𝑥 ∈ (𝑅𝐴) ↔ ∃𝑦𝐴 𝑦𝑅𝑥)
92elima 6085 . . 3 (𝑥 ∈ (𝑅 “ (𝐴 ∩ dom 𝑅)) ↔ ∃𝑦 ∈ (𝐴 ∩ dom 𝑅)𝑦𝑅𝑥)
107, 8, 93bitr4i 303 . 2 (𝑥 ∈ (𝑅𝐴) ↔ 𝑥 ∈ (𝑅 “ (𝐴 ∩ dom 𝑅)))
1110eqriv 2732 1 (𝑅𝐴) = (𝑅 “ (𝐴 ∩ dom 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wa 395   = wceq 1537  wcel 2106  wrex 3068  cin 3962   class class class wbr 5148  dom cdm 5689  cima 5692
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 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702
This theorem is referenced by:  madeval2  27907
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