Theorem resinsn 48745

Description: Restriction to the intersection with a singleton. (Contributed by Zhi Wang, 6-Oct-2025.)

Assertion

Ref	Expression
resinsn	⊢ ((𝐹 ↾ (𝐴 ∩ {𝐵})) = ∅ ↔ ¬ 𝐵 ∈ (dom 𝐹 ∩ 𝐴))

Proof of Theorem resinsn

Step	Hyp	Ref	Expression
1		relres 6021	. . 3 ⊢ Rel (𝐹 ↾ (𝐴 ∩ {𝐵}))
2		reldm0 5936	. . 3 ⊢ (Rel (𝐹 ↾ (𝐴 ∩ {𝐵})) → ((𝐹 ↾ (𝐴 ∩ {𝐵})) = ∅ ↔ dom (𝐹 ↾ (𝐴 ∩ {𝐵})) = ∅))
3	1, 2	ax-mp 5	. 2 ⊢ ((𝐹 ↾ (𝐴 ∩ {𝐵})) = ∅ ↔ dom (𝐹 ↾ (𝐴 ∩ {𝐵})) = ∅)
4		incom 4208	. . . 4 ⊢ ((𝐴 ∩ {𝐵}) ∩ dom 𝐹) = (dom 𝐹 ∩ (𝐴 ∩ {𝐵}))
5		dmres 6028	. . . 4 ⊢ dom (𝐹 ↾ (𝐴 ∩ {𝐵})) = ((𝐴 ∩ {𝐵}) ∩ dom 𝐹)
6		inass 4227	. . . 4 ⊢ ((dom 𝐹 ∩ 𝐴) ∩ {𝐵}) = (dom 𝐹 ∩ (𝐴 ∩ {𝐵}))
7	4, 5, 6	3eqtr4i 2774	. . 3 ⊢ dom (𝐹 ↾ (𝐴 ∩ {𝐵})) = ((dom 𝐹 ∩ 𝐴) ∩ {𝐵})
8	7	eqeq1i 2741	. 2 ⊢ (dom (𝐹 ↾ (𝐴 ∩ {𝐵})) = ∅ ↔ ((dom 𝐹 ∩ 𝐴) ∩ {𝐵}) = ∅)
9		disjsn 4709	. 2 ⊢ (((dom 𝐹 ∩ 𝐴) ∩ {𝐵}) = ∅ ↔ ¬ 𝐵 ∈ (dom 𝐹 ∩ 𝐴))
10	3, 8, 9	3bitri 297	1 ⊢ ((𝐹 ↾ (𝐴 ∩ {𝐵})) = ∅ ↔ ¬ 𝐵 ∈ (dom 𝐹 ∩ 𝐴))

Syntax hints:  ¬ wn 3   ↔ wb 206   = wceq 1540   ∈ wcel 2108   ∩ cin 3949  ∅c0 4332  {csn 4624  dom cdm 5683   ↾ cres 5685  Rel wrel 5688

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 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-sep 5294  ax-nul 5304  ax-pr 5430

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5142  df-opab 5204  df-xp 5689  df-rel 5690  df-dm 5693  df-res 5695

This theorem is referenced by:  tposres2  48753