Theorem ressucdifsn 39686

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 6394	. . . 4 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
2	1	reseq2i 5983	. . 3 ⊢ (𝑅 ↾ suc 𝐴) = (𝑅 ↾ (𝐴 ∪ {𝐴}))
3	2	difeq1i 4062	. 2 ⊢ ((𝑅 ↾ suc 𝐴) ∖ (𝑅 ↾ {𝐴})) = ((𝑅 ↾ (𝐴 ∪ {𝐴})) ∖ (𝑅 ↾ {𝐴}))
4		ressucdifsn2 39685	. 2 ⊢ ((𝑅 ↾ (𝐴 ∪ {𝐴})) ∖ (𝑅 ↾ {𝐴})) = (𝑅 ↾ 𝐴)
5	3, 4	eqtri 2786	1 ⊢ ((𝑅 ↾ suc 𝐴) ∖ (𝑅 ↾ {𝐴})) = (𝑅 ↾ 𝐴)

This proof depends on syntax axioms:   = wceq 1570   ∖ cdif 4048   ∪ cun 4152  {csn 4588   ↾ cres 5660  suc csuc 6390

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  ax-reg 9623

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-opab 5174  df-xp 5664  df-rel 5665  df-res 5670  df-suc 6394

This theorem is used by:  partsuc  40081