Theorem imaindm 6293

Description: The image is unaffected by intersection with the domain. (Contributed by Scott Fenton, 17-Dec-2021.)

Assertion

Ref	Expression
imaindm	⊢ (𝑅 “ 𝐴) = (𝑅 “ (𝐴 ∩ dom 𝑅))

Proof of Theorem imaindm

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3468	. . . . . . 7 ⊢ 𝑦 ∈ V
2		vex 3468	. . . . . . 7 ⊢ 𝑥 ∈ V
3	1, 2	breldm 5893	. . . . . 6 ⊢ (𝑦𝑅𝑥 → 𝑦 ∈ dom 𝑅)
4	3	pm4.71ri 560	. . . . 5 ⊢ (𝑦𝑅𝑥 ↔ (𝑦 ∈ dom 𝑅 ∧ 𝑦𝑅𝑥))
5	4	rexbii 3084	. . . 4 ⊢ (∃𝑦 ∈ 𝐴 𝑦𝑅𝑥 ↔ ∃𝑦 ∈ 𝐴 (𝑦 ∈ dom 𝑅 ∧ 𝑦𝑅𝑥))
6		rexin 4230	. . . 4 ⊢ (∃𝑦 ∈ (𝐴 ∩ dom 𝑅)𝑦𝑅𝑥 ↔ ∃𝑦 ∈ 𝐴 (𝑦 ∈ dom 𝑅 ∧ 𝑦𝑅𝑥))
7	5, 6	bitr4i 278	. . 3 ⊢ (∃𝑦 ∈ 𝐴 𝑦𝑅𝑥 ↔ ∃𝑦 ∈ (𝐴 ∩ dom 𝑅)𝑦𝑅𝑥)
8	2	elima 6057	. . 3 ⊢ (𝑥 ∈ (𝑅 “ 𝐴) ↔ ∃𝑦 ∈ 𝐴 𝑦𝑅𝑥)
9	2	elima 6057	. . 3 ⊢ (𝑥 ∈ (𝑅 “ (𝐴 ∩ dom 𝑅)) ↔ ∃𝑦 ∈ (𝐴 ∩ dom 𝑅)𝑦𝑅𝑥)
10	7, 8, 9	3bitr4i 303	. 2 ⊢ (𝑥 ∈ (𝑅 “ 𝐴) ↔ 𝑥 ∈ (𝑅 “ (𝐴 ∩ dom 𝑅)))
11	10	eqriv 2733	1 ⊢ (𝑅 “ 𝐴) = (𝑅 “ (𝐴 ∩ dom 𝑅))

Syntax hints:   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∃wrex 3061   ∩ cin 3930   class class class wbr 5124  dom cdm 5659   “ cima 5662

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5125  df-opab 5187  df-xp 5665  df-cnv 5667  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672

This theorem is referenced by:  madeval2  27818