Theorem imaindm 32018

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 3354	. . . . . . 7 ⊢ 𝑦 ∈ V
2		vex 3354	. . . . . . 7 ⊢ 𝑥 ∈ V
3	1, 2	breldm 5466	. . . . . 6 ⊢ (𝑦𝑅𝑥 → 𝑦 ∈ dom 𝑅)
4	3	pm4.71ri 550	. . . . 5 ⊢ (𝑦𝑅𝑥 ↔ (𝑦 ∈ dom 𝑅 ∧ 𝑦𝑅𝑥))
5	4	rexbii 3189	. . . 4 ⊢ (∃𝑦 ∈ 𝐴 𝑦𝑅𝑥 ↔ ∃𝑦 ∈ 𝐴 (𝑦 ∈ dom 𝑅 ∧ 𝑦𝑅𝑥))
6		elin 3947	. . . . . . 7 ⊢ (𝑦 ∈ (𝐴 ∩ dom 𝑅) ↔ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ dom 𝑅))
7	6	anbi1i 610	. . . . . 6 ⊢ ((𝑦 ∈ (𝐴 ∩ dom 𝑅) ∧ 𝑦𝑅𝑥) ↔ ((𝑦 ∈ 𝐴 ∧ 𝑦 ∈ dom 𝑅) ∧ 𝑦𝑅𝑥))
8		anass 454	. . . . . 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 3187	. . . 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 5611	. . 3 ⊢ (𝑥 ∈ (𝑅 “ 𝐴) ↔ ∃𝑦 ∈ 𝐴 𝑦𝑅𝑥)
13	2	elima 5611	. . 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 2768	1 ⊢ (𝑅 “ 𝐴) = (𝑅 “ (𝐴 ∩ dom 𝑅))

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