Theorem ressucdifsn2 39610

Description: The difference between restrictions to the successor and the singleton of a class is the restriction to the class, see ressucdifsn 39611. (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 9628	. 2 ⊢ (𝐴 ∩ {𝐴}) = ∅
2		disjresundif 39369	. 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 4048   ∩ cin 4092   ∪ cun 4152  ∅c0 4281  {csn 4588   ↾ cres 5657

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5252  ax-pr 5395  ax-reg 9610

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-opab 5171  df-xp 5661  df-rel 5662  df-res 5667

This theorem is used by:  ressucdifsn  39611  partsuc2  40005