Theorem resinsn 48876

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 5960	. . 3 ⊢ Rel (𝐹 ↾ (𝐴 ∩ {𝐵}))
2		reldm0 5874	. . 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 4162	. . . 4 ⊢ ((𝐴 ∩ {𝐵}) ∩ dom 𝐹) = (dom 𝐹 ∩ (𝐴 ∩ {𝐵}))
5		dmres 5967	. . . 4 ⊢ dom (𝐹 ↾ (𝐴 ∩ {𝐵})) = ((𝐴 ∩ {𝐵}) ∩ dom 𝐹)
6		inass 4181	. . . 4 ⊢ ((dom 𝐹 ∩ 𝐴) ∩ {𝐵}) = (dom 𝐹 ∩ (𝐴 ∩ {𝐵}))
7	4, 5, 6	3eqtr4i 2762	. . 3 ⊢ dom (𝐹 ↾ (𝐴 ∩ {𝐵})) = ((dom 𝐹 ∩ 𝐴) ∩ {𝐵})
8	7	eqeq1i 2734	. 2 ⊢ (dom (𝐹 ↾ (𝐴 ∩ {𝐵})) = ∅ ↔ ((dom 𝐹 ∩ 𝐴) ∩ {𝐵}) = ∅)
9		disjsn 4665	. 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 2109   ∩ cin 3904  ∅c0 4286  {csn 4579  dom cdm 5623   ↾ cres 5625  Rel wrel 5628

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-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

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-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5096  df-opab 5158  df-xp 5629  df-rel 5630  df-dm 5633  df-res 5635

This theorem is referenced by:  tposres2  48884