Theorem ressucdifsn2 39339

Description: The difference between restrictions to the successor and the singleton of a class is the restriction to the class, see ressucdifsn 39340. (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 9582	. 2 ⊢ (𝐴 ∩ {𝐴}) = ∅
2		disjresundif 39098	. 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 1570   ∖ cdif 3895   ∪ cun 3896   ∩ cin 3897  ∅c0 4278  {csn 4583   ↾ cres 5649

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5248  ax-pr 5390  ax-reg 9564

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 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-opab 5167  df-xp 5653  df-rel 5654  df-res 5659

This theorem is used by:  ressucdifsn  39340  partsuc2  39734