Theorem resdmdfsn 6033

Description: Restricting a class to its domain without a set is the same as restricting the class to the universe without this set. (Contributed by AV, 2-Dec-2018.) Remove antecedent. (Revised by Eric Schmidt, 16-Jun-2026.)

Assertion

Ref	Expression
resdmdfsn	⊢ (𝑅 ↾ (V ∖ {𝑋})) = (𝑅 ↾ (dom 𝑅 ∖ {𝑋}))

Proof of Theorem resdmdfsn

Step	Hyp	Ref	Expression
1		resindm 6031	. 2 ⊢ (𝑅 ↾ ((V ∖ {𝑋}) ∩ dom 𝑅)) = (𝑅 ↾ (V ∖ {𝑋}))
2		indif1 4235	. . . 4 ⊢ ((V ∖ {𝑋}) ∩ dom 𝑅) = ((V ∩ dom 𝑅) ∖ {𝑋})
3		inv1 4355	. . . . . 6 ⊢ (dom 𝑅 ∩ V) = dom 𝑅
4	3	ineqcomi 4164	. . . . 5 ⊢ (V ∩ dom 𝑅) = dom 𝑅
5	4	difeq1i 4077	. . . 4 ⊢ ((V ∩ dom 𝑅) ∖ {𝑋}) = (dom 𝑅 ∖ {𝑋})
6	2, 5	eqtri 2788	. . 3 ⊢ ((V ∖ {𝑋}) ∩ dom 𝑅) = (dom 𝑅 ∖ {𝑋})
7	6	reseq2i 5977	. 2 ⊢ (𝑅 ↾ ((V ∖ {𝑋}) ∩ dom 𝑅)) = (𝑅 ↾ (dom 𝑅 ∖ {𝑋}))
8	1, 7	eqtr3i 2790	1 ⊢ (𝑅 ↾ (V ∖ {𝑋})) = (𝑅 ↾ (dom 𝑅 ∖ {𝑋}))

This proof depends on syntax axioms:   = wceq 1570  Vcvv 3457   ∖ cdif 3903   ∩ cin 3905  {csn 4591  dom cdm 5663   ↾ cres 5665

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:  funresdfunsn  7193  fresunsn  33043