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Theorem imaindm 32018
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 3354 . . . . . . 7 𝑦 ∈ V
2 vex 3354 . . . . . . 7 𝑥 ∈ V
31, 2breldm 5466 . . . . . 6 (𝑦𝑅𝑥𝑦 ∈ dom 𝑅)
43pm4.71ri 550 . . . . 5 (𝑦𝑅𝑥 ↔ (𝑦 ∈ dom 𝑅𝑦𝑅𝑥))
54rexbii 3189 . . . 4 (∃𝑦𝐴 𝑦𝑅𝑥 ↔ ∃𝑦𝐴 (𝑦 ∈ dom 𝑅𝑦𝑅𝑥))
6 elin 3947 . . . . . . 7 (𝑦 ∈ (𝐴 ∩ dom 𝑅) ↔ (𝑦𝐴𝑦 ∈ dom 𝑅))
76anbi1i 610 . . . . . 6 ((𝑦 ∈ (𝐴 ∩ dom 𝑅) ∧ 𝑦𝑅𝑥) ↔ ((𝑦𝐴𝑦 ∈ dom 𝑅) ∧ 𝑦𝑅𝑥))
8 anass 454 . . . . . 6 (((𝑦𝐴𝑦 ∈ dom 𝑅) ∧ 𝑦𝑅𝑥) ↔ (𝑦𝐴 ∧ (𝑦 ∈ dom 𝑅𝑦𝑅𝑥)))
97, 8bitri 264 . . . . 5 ((𝑦 ∈ (𝐴 ∩ dom 𝑅) ∧ 𝑦𝑅𝑥) ↔ (𝑦𝐴 ∧ (𝑦 ∈ dom 𝑅𝑦𝑅𝑥)))
109rexbii2 3187 . . . 4 (∃𝑦 ∈ (𝐴 ∩ dom 𝑅)𝑦𝑅𝑥 ↔ ∃𝑦𝐴 (𝑦 ∈ dom 𝑅𝑦𝑅𝑥))
115, 10bitr4i 267 . . 3 (∃𝑦𝐴 𝑦𝑅𝑥 ↔ ∃𝑦 ∈ (𝐴 ∩ dom 𝑅)𝑦𝑅𝑥)
122elima 5611 . . 3 (𝑥 ∈ (𝑅𝐴) ↔ ∃𝑦𝐴 𝑦𝑅𝑥)
132elima 5611 . . 3 (𝑥 ∈ (𝑅 “ (𝐴 ∩ dom 𝑅)) ↔ ∃𝑦 ∈ (𝐴 ∩ dom 𝑅)𝑦𝑅𝑥)
1411, 12, 133bitr4i 292 . 2 (𝑥 ∈ (𝑅𝐴) ↔ 𝑥 ∈ (𝑅 “ (𝐴 ∩ dom 𝑅)))
1514eqriv 2768 1 (𝑅𝐴) = (𝑅 “ (𝐴 ∩ dom 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wa 382   = wceq 1631  wcel 2145  wrex 3062  cin 3722   class class class wbr 4787  dom cdm 5250  cima 5253
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-br 4788  df-opab 4848  df-xp 5256  df-cnv 5258  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263
This theorem is referenced by:  madeval2  32273
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