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Theorem imaindm 6308
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 3477 . . . . . . 7 𝑦 ∈ V
2 vex 3477 . . . . . . 7 𝑥 ∈ V
31, 2breldm 5915 . . . . . 6 (𝑦𝑅𝑥𝑦 ∈ dom 𝑅)
43pm4.71ri 559 . . . . 5 (𝑦𝑅𝑥 ↔ (𝑦 ∈ dom 𝑅𝑦𝑅𝑥))
54rexbii 3091 . . . 4 (∃𝑦𝐴 𝑦𝑅𝑥 ↔ ∃𝑦𝐴 (𝑦 ∈ dom 𝑅𝑦𝑅𝑥))
6 rexin 4242 . . . 4 (∃𝑦 ∈ (𝐴 ∩ dom 𝑅)𝑦𝑅𝑥 ↔ ∃𝑦𝐴 (𝑦 ∈ dom 𝑅𝑦𝑅𝑥))
75, 6bitr4i 277 . . 3 (∃𝑦𝐴 𝑦𝑅𝑥 ↔ ∃𝑦 ∈ (𝐴 ∩ dom 𝑅)𝑦𝑅𝑥)
82elima 6073 . . 3 (𝑥 ∈ (𝑅𝐴) ↔ ∃𝑦𝐴 𝑦𝑅𝑥)
92elima 6073 . . 3 (𝑥 ∈ (𝑅 “ (𝐴 ∩ dom 𝑅)) ↔ ∃𝑦 ∈ (𝐴 ∩ dom 𝑅)𝑦𝑅𝑥)
107, 8, 93bitr4i 302 . 2 (𝑥 ∈ (𝑅𝐴) ↔ 𝑥 ∈ (𝑅 “ (𝐴 ∩ dom 𝑅)))
1110eqriv 2725 1 (𝑅𝐴) = (𝑅 “ (𝐴 ∩ dom 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wa 394   = wceq 1533  wcel 2098  wrex 3067  cin 3948   class class class wbr 5152  dom cdm 5682  cima 5685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-br 5153  df-opab 5215  df-xp 5688  df-cnv 5690  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695
This theorem is referenced by:  madeval2  27800
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