Theorem ressucdifsn2 39420

Description: The difference between restrictions to the successor and the singleton of a class is the restriction to the class, see ressucdifsn 39421. (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 9604	. 2 ⊢ (𝐴 ∩ {𝐴}) = ∅
2		disjresundif 39179	. 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 4045   ∩ cin 4089   ∪ cun 4149  ∅c0 4278  {csn 4583   ↾ cres 5652

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390  ax-reg 9586

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-opab 5167  df-xp 5656  df-rel 5657  df-res 5662

This theorem is used by:  ressucdifsn  39421  partsuc2  39815