Theorem resinsn 49359

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 5964	. . 3 ⊢ Rel (𝐹 ↾ (𝐴 ∩ {𝐵}))
2		reldm0 5877	. . 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 4150	. . . 4 ⊢ ((𝐴 ∩ {𝐵}) ∩ dom 𝐹) = (dom 𝐹 ∩ (𝐴 ∩ {𝐵}))
5		dmres 5971	. . . 4 ⊢ dom (𝐹 ↾ (𝐴 ∩ {𝐵})) = ((𝐴 ∩ {𝐵}) ∩ dom 𝐹)
6		inass 4169	. . . 4 ⊢ ((dom 𝐹 ∩ 𝐴) ∩ {𝐵}) = (dom 𝐹 ∩ (𝐴 ∩ {𝐵}))
7	4, 5, 6	3eqtr4i 2770	. . 3 ⊢ dom (𝐹 ↾ (𝐴 ∩ {𝐵})) = ((dom 𝐹 ∩ 𝐴) ∩ {𝐵})
8	7	eqeq1i 2742	. 2 ⊢ (dom (𝐹 ↾ (𝐴 ∩ {𝐵})) = ∅ ↔ ((dom 𝐹 ∩ 𝐴) ∩ {𝐵}) = ∅)
9		disjsn 4656	. 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 1542   ∈ wcel 2114   ∩ cin 3889  ∅c0 4274  {csn 4568  dom cdm 5624   ↾ cres 5626  Rel wrel 5629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5231  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-xp 5630  df-rel 5631  df-dm 5634  df-res 5636

This theorem is referenced by:  tposres2  49367