Theorem resindm 6037

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 6020	. . 3 ⊢ dom (𝐴 ↾ 𝐵) = (𝐵 ∩ dom 𝐴)
2	1	reseq2i 5983	. 2 ⊢ ((𝐴 ↾ 𝐵) ↾ dom (𝐴 ↾ 𝐵)) = ((𝐴 ↾ 𝐵) ↾ (𝐵 ∩ dom 𝐴))
3		relres 6012	. . 3 ⊢ Rel (𝐴 ↾ 𝐵)
4		resdm 6033	. . 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 4136	. . 3 ⊢ (𝐵 ∩ dom 𝐴) ⊆ 𝐵
7	6	resabs1i 6015	. 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 1570   ∩ cin 4092  dom cdm 5658   ↾ cres 5660  Rel wrel 5663

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-xp 5664  df-rel 5665  df-dm 5668  df-res 5670

This theorem is used by:  resdmdfsn  6039  imadifssranOLD  6221  resfnfinfin  9364  resfifsupp  9426  coidmid  38225  nvo3  38230  poimirlem3  38809  fresin2  46460  limsupvaluz  46991  cncfuni  47169  fourierdlem48  47437  fourierdlem49  47438  fourierdlem113  47502  sssmf  48021