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Theorem imaindm 31806
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 3234 . . . . . . 7 𝑦 ∈ V
2 vex 3234 . . . . . . 7 𝑥 ∈ V
31, 2breldm 5361 . . . . . 6 (𝑦𝑅𝑥𝑦 ∈ dom 𝑅)
43pm4.71ri 666 . . . . 5 (𝑦𝑅𝑥 ↔ (𝑦 ∈ dom 𝑅𝑦𝑅𝑥))
54rexbii 3070 . . . 4 (∃𝑦𝐴 𝑦𝑅𝑥 ↔ ∃𝑦𝐴 (𝑦 ∈ dom 𝑅𝑦𝑅𝑥))
6 elin 3829 . . . . . . 7 (𝑦 ∈ (𝐴 ∩ dom 𝑅) ↔ (𝑦𝐴𝑦 ∈ dom 𝑅))
76anbi1i 731 . . . . . 6 ((𝑦 ∈ (𝐴 ∩ dom 𝑅) ∧ 𝑦𝑅𝑥) ↔ ((𝑦𝐴𝑦 ∈ dom 𝑅) ∧ 𝑦𝑅𝑥))
8 anass 682 . . . . . 6 (((𝑦𝐴𝑦 ∈ dom 𝑅) ∧ 𝑦𝑅𝑥) ↔ (𝑦𝐴 ∧ (𝑦 ∈ dom 𝑅𝑦𝑅𝑥)))
97, 8bitri 264 . . . . 5 ((𝑦 ∈ (𝐴 ∩ dom 𝑅) ∧ 𝑦𝑅𝑥) ↔ (𝑦𝐴 ∧ (𝑦 ∈ dom 𝑅𝑦𝑅𝑥)))
109rexbii2 3068 . . . 4 (∃𝑦 ∈ (𝐴 ∩ dom 𝑅)𝑦𝑅𝑥 ↔ ∃𝑦𝐴 (𝑦 ∈ dom 𝑅𝑦𝑅𝑥))
115, 10bitr4i 267 . . 3 (∃𝑦𝐴 𝑦𝑅𝑥 ↔ ∃𝑦 ∈ (𝐴 ∩ dom 𝑅)𝑦𝑅𝑥)
122elima 5506 . . 3 (𝑥 ∈ (𝑅𝐴) ↔ ∃𝑦𝐴 𝑦𝑅𝑥)
132elima 5506 . . 3 (𝑥 ∈ (𝑅 “ (𝐴 ∩ dom 𝑅)) ↔ ∃𝑦 ∈ (𝐴 ∩ dom 𝑅)𝑦𝑅𝑥)
1411, 12, 133bitr4i 292 . 2 (𝑥 ∈ (𝑅𝐴) ↔ 𝑥 ∈ (𝑅 “ (𝐴 ∩ dom 𝑅)))
1514eqriv 2648 1 (𝑅𝐴) = (𝑅 “ (𝐴 ∩ dom 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wa 383   = wceq 1523  wcel 2030  wrex 2942  cin 3606   class class class wbr 4685  dom cdm 5143  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-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  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-xp 5149  df-cnv 5151  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156
This theorem is referenced by:  madeval2  32061
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