Theorem imaindm 31682

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 3203	. . . . . . 7 ⊢ 𝑦 ∈ V
2		vex 3203	. . . . . . 7 ⊢ 𝑥 ∈ V
3	1, 2	breldm 5329	. . . . . 6 ⊢ (𝑦𝑅𝑥 → 𝑦 ∈ dom 𝑅)
4	3	pm4.71ri 665	. . . . 5 ⊢ (𝑦𝑅𝑥 ↔ (𝑦 ∈ dom 𝑅 ∧ 𝑦𝑅𝑥))
5	4	rexbii 3041	. . . 4 ⊢ (∃𝑦 ∈ 𝐴 𝑦𝑅𝑥 ↔ ∃𝑦 ∈ 𝐴 (𝑦 ∈ dom 𝑅 ∧ 𝑦𝑅𝑥))
6		elin 3796	. . . . . . 7 ⊢ (𝑦 ∈ (𝐴 ∩ dom 𝑅) ↔ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ dom 𝑅))
7	6	anbi1i 731	. . . . . 6 ⊢ ((𝑦 ∈ (𝐴 ∩ dom 𝑅) ∧ 𝑦𝑅𝑥) ↔ ((𝑦 ∈ 𝐴 ∧ 𝑦 ∈ dom 𝑅) ∧ 𝑦𝑅𝑥))
8		anass 681	. . . . . 6 ⊢ (((𝑦 ∈ 𝐴 ∧ 𝑦 ∈ dom 𝑅) ∧ 𝑦𝑅𝑥) ↔ (𝑦 ∈ 𝐴 ∧ (𝑦 ∈ dom 𝑅 ∧ 𝑦𝑅𝑥)))
9	7, 8	bitri 264	. . . . 5 ⊢ ((𝑦 ∈ (𝐴 ∩ dom 𝑅) ∧ 𝑦𝑅𝑥) ↔ (𝑦 ∈ 𝐴 ∧ (𝑦 ∈ dom 𝑅 ∧ 𝑦𝑅𝑥)))
10	9	rexbii2 3039	. . . 4 ⊢ (∃𝑦 ∈ (𝐴 ∩ dom 𝑅)𝑦𝑅𝑥 ↔ ∃𝑦 ∈ 𝐴 (𝑦 ∈ dom 𝑅 ∧ 𝑦𝑅𝑥))
11	5, 10	bitr4i 267	. . 3 ⊢ (∃𝑦 ∈ 𝐴 𝑦𝑅𝑥 ↔ ∃𝑦 ∈ (𝐴 ∩ dom 𝑅)𝑦𝑅𝑥)
12	2	elima 5471	. . 3 ⊢ (𝑥 ∈ (𝑅 “ 𝐴) ↔ ∃𝑦 ∈ 𝐴 𝑦𝑅𝑥)
13	2	elima 5471	. . 3 ⊢ (𝑥 ∈ (𝑅 “ (𝐴 ∩ dom 𝑅)) ↔ ∃𝑦 ∈ (𝐴 ∩ dom 𝑅)𝑦𝑅𝑥)
14	11, 12, 13	3bitr4i 292	. 2 ⊢ (𝑥 ∈ (𝑅 “ 𝐴) ↔ 𝑥 ∈ (𝑅 “ (𝐴 ∩ dom 𝑅)))
15	14	eqriv 2619	1 ⊢ (𝑅 “ 𝐴) = (𝑅 “ (𝐴 ∩ dom 𝑅))

Syntax hints:   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∃wrex 2913   ∩ cin 3573   class class class wbr 4653  dom cdm 5114   “ cima 5117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127

This theorem is referenced by:  madeval2  31936