Theorem resindm 6023

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 6005	. . 3 ⊢ dom (𝐴 ↾ 𝐵) = (𝐵 ∩ dom 𝐴)
2	1	reseq2i 5969	. 2 ⊢ ((𝐴 ↾ 𝐵) ↾ dom (𝐴 ↾ 𝐵)) = ((𝐴 ↾ 𝐵) ↾ (𝐵 ∩ dom 𝐴))
3		relres 5998	. . 3 ⊢ Rel (𝐴 ↾ 𝐵)
4		resdm 6019	. . 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 4182	. . 3 ⊢ (𝐵 ∩ dom 𝐴) ⊆ 𝐵
7	6	resabs1i 6000	. 2 ⊢ ((𝐴 ↾ 𝐵) ↾ (𝐵 ∩ dom 𝐴)) = (𝐴 ↾ (𝐵 ∩ dom 𝐴))
8	2, 5, 7	3eqtr3ri 2792	1 ⊢ (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = (𝐴 ↾ 𝐵)

This proof depends on syntax axioms:   = wceq 1570   ∩ cin 3898  dom cdm 5655   ↾ cres 5657  Rel wrel 5660

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5661  df-rel 5662  df-dm 5665  df-res 5667

This theorem is used by:  resdmdfsn  6025  imadifssran  6197  resfnfinfin  9304  resfifsupp  9367  poimirlem3  38372  fresin2  46004  limsupvaluz  46536  cncfuni  46714  fourierdlem48  46982  fourierdlem49  46983  fourierdlem113  47047  sssmf  47566