Theorem ressucdifsn2 38225

Description: The difference between restrictions to the successor and the singleton of a class is the restriction to the class, see ressucdifsn 38226. (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 9533	. 2 ⊢ (𝐴 ∩ {𝐴}) = ∅
2		disjresundif 38224	. 2 ⊢ ((𝐴 ∩ {𝐴}) = ∅ → ((𝑅 ↾ (𝐴 ∪ {𝐴})) ∖ (𝑅 ↾ {𝐴})) = (𝑅 ↾ 𝐴))
3	1, 2	ax-mp 5	1 ⊢ ((𝑅 ↾ (𝐴 ∪ {𝐴})) ∖ (𝑅 ↾ {𝐴})) = (𝑅 ↾ 𝐴)

Syntax hints:   = wceq 1540   ∖ cdif 3908   ∪ cun 3909   ∩ cin 3910  ∅c0 4292  {csn 4585   ↾ cres 5633

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382  ax-reg 9521

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-opab 5165  df-xp 5637  df-rel 5638  df-res 5643

This theorem is referenced by:  ressucdifsn  38226  partsuc2  38764