Theorem br1cnvssrres 39089

Description: Restricted converse subset binary relation. (Contributed by Peter Mazsa, 25-Nov-2019.)

Assertion

Ref	Expression
br1cnvssrres	⊢ (𝐵 ∈ 𝑉 → (𝐵◡( S ↾ 𝐴)𝐶 ↔ (𝐶 ∈ 𝐴 ∧ 𝐶 ⊆ 𝐵)))

Proof of Theorem br1cnvssrres

Step	Hyp	Ref	Expression
1		relres 5991	. . 3 ⊢ Rel ( S ↾ 𝐴)
2	1	relbrcnv 6096	. 2 ⊢ (𝐵◡( S ↾ 𝐴)𝐶 ↔ 𝐶( S ↾ 𝐴)𝐵)
3		brssrres 39088	. 2 ⊢ (𝐵 ∈ 𝑉 → (𝐶( S ↾ 𝐴)𝐵 ↔ (𝐶 ∈ 𝐴 ∧ 𝐶 ⊆ 𝐵)))
4	2, 3	bitrid 285	1 ⊢ (𝐵 ∈ 𝑉 → (𝐵◡( S ↾ 𝐴)𝐶 ↔ (𝐶 ∈ 𝐴 ∧ 𝐶 ⊆ 𝐵)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   ∈ wcel 2143   ⊆ wss 3905   class class class wbr 5101  ^◡ccnv 5647   ↾ cres 5650   S cssr 38690

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5247  ax-pr 5391

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-sb 2092  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5102  df-opab 5164  df-xp 5654  df-rel 5655  df-cnv 5656  df-res 5660  df-ssr 39082

This theorem is referenced by: (None)