Theorem ressucdifsn 38733

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 6331	. . . 4 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
2	1	reseq2i 5943	. . 3 ⊢ (𝑅 ↾ suc 𝐴) = (𝑅 ↾ (𝐴 ∪ {𝐴}))
3	2	difeq1i 4076	. 2 ⊢ ((𝑅 ↾ suc 𝐴) ∖ (𝑅 ↾ {𝐴})) = ((𝑅 ↾ (𝐴 ∪ {𝐴})) ∖ (𝑅 ↾ {𝐴}))
4		ressucdifsn2 38732	. 2 ⊢ ((𝑅 ↾ (𝐴 ∪ {𝐴})) ∖ (𝑅 ↾ {𝐴})) = (𝑅 ↾ 𝐴)
5	3, 4	eqtri 2760	1 ⊢ ((𝑅 ↾ suc 𝐴) ∖ (𝑅 ↾ {𝐴})) = (𝑅 ↾ 𝐴)

Syntax hints:   = wceq 1542   ∖ cdif 3900   ∪ cun 3901  {csn 4582   ↾ cres 5634  suc csuc 6327

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-pr 5379  ax-reg 9509

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-opab 5163  df-xp 5638  df-rel 5639  df-res 5644  df-suc 6331

This theorem is referenced by:  partsuc  39128