Theorem ressucdifsn 38823

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 6323	. . . 4 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
2	1	reseq2i 5935	. . 3 ⊢ (𝑅 ↾ suc 𝐴) = (𝑅 ↾ (𝐴 ∪ {𝐴}))
3	2	difeq1i 4063	. 2 ⊢ ((𝑅 ↾ suc 𝐴) ∖ (𝑅 ↾ {𝐴})) = ((𝑅 ↾ (𝐴 ∪ {𝐴})) ∖ (𝑅 ↾ {𝐴}))
4		ressucdifsn2 38822	. 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 3887   ∪ cun 3888  {csn 4568   ↾ cres 5626  suc csuc 6319

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 5231  ax-pr 5370  ax-reg 9500

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 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-opab 5149  df-xp 5630  df-rel 5631  df-res 5636  df-suc 6323

This theorem is referenced by:  partsuc  39218