Theorem ressucdifsn2 39164

Description: The difference between restrictions to the successor and the singleton of a class is the restriction to the class, see ressucdifsn 39165. (Contributed by Peter Mazsa, 24-Jul-2024.)

Assertion

Ref	Expression
ressucdifsn2	⊢ ((𝑅 ↾ (𝐴 ∪ {𝐴})) ∖ (𝑅 ↾ {𝐴})) = (𝑅 ↾ 𝐴)

Proof of Theorem ressucdifsn2

Step	Hyp	Ref	Expression
1		disjcsn 9570	. 2 ⊢ (𝐴 ∩ {𝐴}) = ∅
2		disjresundif 38923	. 2 ⊢ ((𝐴 ∩ {𝐴}) = ∅ → ((𝑅 ↾ (𝐴 ∪ {𝐴})) ∖ (𝑅 ↾ {𝐴})) = (𝑅 ↾ 𝐴))
3	1, 2	ax-mp 5	1 ⊢ ((𝑅 ↾ (𝐴 ∪ {𝐴})) ∖ (𝑅 ↾ {𝐴})) = (𝑅 ↾ 𝐴)

This proof depends on syntax axioms:   = wceq 1569   ∖ cdif 3901   ∪ cun 3902   ∩ cin 3903  ∅c0 4285  {csn 4588   ↾ cres 5662

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5256  ax-pr 5403  ax-reg 9552

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-opab 5173  df-xp 5666  df-rel 5667  df-res 5672

This theorem is used by:  ressucdifsn  39165  partsuc2  39559