Theorem ressucdifsn 39118

Description: The difference between restrictions to the successor and the singleton of a class is the restriction to the class. (Contributed by Peter Mazsa, 20-Sep-2024.)

Assertion

Ref	Expression
ressucdifsn	⊢ ((𝑅 ↾ suc 𝐴) ∖ (𝑅 ↾ {𝐴})) = (𝑅 ↾ 𝐴)

Proof of Theorem ressucdifsn

Step	Hyp	Ref	Expression
1		df-suc 6368	. . . 4 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
2	1	reseq2i 5977	. . 3 ⊢ (𝑅 ↾ suc 𝐴) = (𝑅 ↾ (𝐴 ∪ {𝐴}))
3	2	difeq1i 4078	. 2 ⊢ ((𝑅 ↾ suc 𝐴) ∖ (𝑅 ↾ {𝐴})) = ((𝑅 ↾ (𝐴 ∪ {𝐴})) ∖ (𝑅 ↾ {𝐴}))
4		ressucdifsn2 39117	. 2 ⊢ ((𝑅 ↾ (𝐴 ∪ {𝐴})) ∖ (𝑅 ↾ {𝐴})) = (𝑅 ↾ 𝐴)
5	3, 4	eqtri 2786	1 ⊢ ((𝑅 ↾ suc 𝐴) ∖ (𝑅 ↾ {𝐴})) = (𝑅 ↾ 𝐴)

Syntax hints:   = wceq 1570   ∖ cdif 3903   ∪ cun 3904  {csn 4590   ↾ cres 5665  suc csuc 6364

This theorem was proved from 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 5258  ax-pr 5406  ax-reg 9555

This theorem 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 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-opab 5175  df-xp 5669  df-rel 5670  df-res 5675  df-suc 6368

This theorem is referenced by:  partsuc  39513