Theorem resdmdfsnOLD 6032

Description: Obsolete version of resdmdfsn 6031 as of 16-Jun-2026. (Contributed by AV, 2-Dec-2018.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
resdmdfsnOLD	⊢ (Rel 𝑅 → (𝑅 ↾ (V ∖ {𝑋})) = (𝑅 ↾ (dom 𝑅 ∖ {𝑋})))

Proof of Theorem resdmdfsnOLD

Step	Hyp	Ref	Expression
1		resindmOLD 6030	. 2 ⊢ (Rel 𝑅 → (𝑅 ↾ ((V ∖ {𝑋}) ∩ dom 𝑅)) = (𝑅 ↾ (V ∖ {𝑋})))
2		indif1 4235	. . . 4 ⊢ ((V ∖ {𝑋}) ∩ dom 𝑅) = ((V ∩ dom 𝑅) ∖ {𝑋})
3		incom 4162	. . . . . 6 ⊢ (V ∩ dom 𝑅) = (dom 𝑅 ∩ V)
4		inv1 4355	. . . . . 6 ⊢ (dom 𝑅 ∩ V) = dom 𝑅
5	3, 4	eqtri 2786	. . . . 5 ⊢ (V ∩ dom 𝑅) = dom 𝑅
6	5	difeq1i 4077	. . . 4 ⊢ ((V ∩ dom 𝑅) ∖ {𝑋}) = (dom 𝑅 ∖ {𝑋})
7	2, 6	eqtri 2786	. . 3 ⊢ ((V ∖ {𝑋}) ∩ dom 𝑅) = (dom 𝑅 ∖ {𝑋})
8	7	reseq2i 5975	. 2 ⊢ (𝑅 ↾ ((V ∖ {𝑋}) ∩ dom 𝑅)) = (𝑅 ↾ (dom 𝑅 ∖ {𝑋}))
9	1, 8	eqtr3di 2813	1 ⊢ (Rel 𝑅 → (𝑅 ↾ (V ∖ {𝑋})) = (𝑅 ↾ (dom 𝑅 ∖ {𝑋})))

This proof depends on syntax axioms:   → wi 4   = wceq 1570  Vcvv 3455   ∖ cdif 3902   ∩ cin 3904  {csn 4589  dom cdm 5661   ↾ cres 5663  Rel wrel 5666

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pr 5404

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-xp 5667  df-rel 5668  df-dm 5671  df-res 5673

This theorem is used by: (None)