Theorem resindm 6031

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 6013	. . 3 ⊢ dom (𝐴 ↾ 𝐵) = (𝐵 ∩ dom 𝐴)
2	1	reseq2i 5977	. 2 ⊢ ((𝐴 ↾ 𝐵) ↾ dom (𝐴 ↾ 𝐵)) = ((𝐴 ↾ 𝐵) ↾ (𝐵 ∩ dom 𝐴))
3		relres 6006	. . 3 ⊢ Rel (𝐴 ↾ 𝐵)
4		resdm 6027	. . 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 4189	. . 3 ⊢ (𝐵 ∩ dom 𝐴) ⊆ 𝐵
7	6	resabs1i 6008	. 2 ⊢ ((𝐴 ↾ 𝐵) ↾ (𝐵 ∩ dom 𝐴)) = (𝐴 ↾ (𝐵 ∩ dom 𝐴))
8	2, 5, 7	3eqtr3ri 2797	1 ⊢ (𝐴 ↾ (𝐵 ∩ dom 𝐴)) = (𝐴 ↾ 𝐵)

This proof depends on syntax axioms:   = wceq 1570   ∩ cin 3905  dom cdm 5663   ↾ cres 5665  Rel wrel 5668

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 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

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 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-xp 5669  df-rel 5670  df-dm 5673  df-res 5675

This theorem is used by:  resdmdfsn  6033  imadifssran  6204  resfnfinfin  9301  resfifsupp  9364  poimirlem3  38331  fresin2  45948  limsupvaluz  46480  cncfuni  46658  fourierdlem48  46926  fourierdlem49  46927  fourierdlem113  46991  sssmf  47510