Theorem resindm 6024

Description: When restricting a class, intersecting with the domain of the class has no effect. (Contributed by FL, 6-Oct-2008.) Remove antecedent. (Revised by Eric Schmidt, 16-Jun-2026.)

Assertion

Ref	Expression
resindm	⊢ (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = (𝐴 ↾ 𝐵)

Proof of Theorem resindm

Step	Hyp	Ref	Expression
1		dmres 6008	. . 3 ⊢ dom (𝐴 ↾ 𝐵) = (𝐵 ∩ dom 𝐴)
2	1	reseq2i 5971	. 2 ⊢ ((𝐴 ↾ 𝐵) ↾ dom (𝐴 ↾ 𝐵)) = ((𝐴 ↾ 𝐵) ↾ (𝐵 ∩ dom 𝐴))
3		relres 6000	. . 3 ⊢ Rel (𝐴 ↾ 𝐵)
4		resdm 6020	. . 3 ⊢ (Rel (𝐴 ↾ 𝐵) → ((𝐴 ↾ 𝐵) ↾ dom (𝐴 ↾ 𝐵)) = (𝐴 ↾ 𝐵))
5	3, 4	ax-mp 5	. 2 ⊢ ((𝐴 ↾ 𝐵) ↾ dom (𝐴 ↾ 𝐵)) = (𝐴 ↾ 𝐵)
6		inss1 4135	. . 3 ⊢ (𝐵 ∩ dom 𝐴) ⊆ 𝐵
7	6	resabs1i 6003	. 2 ⊢ ((𝐴 ↾ 𝐵) ↾ (𝐵 ∩ dom 𝐴)) = (𝐴 ↾ (𝐵 ∩ dom 𝐴))
8	2, 5, 7	3eqtr3ri 2795	1 ⊢ (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = (𝐴 ↾ 𝐵)

This proof depends on syntax axioms:   = wceq 1569   ∩ cin 4091  dom cdm 5654   ↾ cres 5656  Rel wrel 5659

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735  ax-sep 5251  ax-pr 5394

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3416  df-v 3455  df-ss 3900  df-dif 4049  df-in 4093  df-un 4153  df-nul 4281  df-if 4485  df-sn 4588  df-pr 4590  df-op 4838  df-br 5106  df-opab 5170  df-xp 5660  df-rel 5661  df-dm 5664  df-res 5666

This theorem is used by:  resdmdfsn  6026  imadifssranOLD  6208  resfnfinfin  9337  resfifsupp  9399  poimirlem3  38681  fresin2  46326  limsupvaluz  46857  cncfuni  47035  fourierdlem48  47303  fourierdlem49  47304  fourierdlem113  47368  sssmf  47887