Theorem resinsnALT 48834

Description: Restriction to the intersection with a singleton. (Contributed by Zhi Wang, 6-Oct-2025.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

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

Proof of Theorem resinsnALT

Step	Hyp	Ref	Expression
1		relres 5965	. . 3 ⊢ Rel (𝐹 ↾ (𝐴 ∩ {𝐵}))
2		reldm0 5881	. . 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		dmres 5972	. . . 4 ⊢ dom (𝐹 ↾ (𝐴 ∩ {𝐵})) = ((𝐴 ∩ {𝐵}) ∩ dom 𝐹)
5		incom 4168	. . . 4 ⊢ ((𝐴 ∩ {𝐵}) ∩ dom 𝐹) = (dom 𝐹 ∩ (𝐴 ∩ {𝐵}))
6	4, 5	eqtri 2752	. . 3 ⊢ dom (𝐹 ↾ (𝐴 ∩ {𝐵})) = (dom 𝐹 ∩ (𝐴 ∩ {𝐵}))
7	6	eqeq1i 2734	. 2 ⊢ (dom (𝐹 ↾ (𝐴 ∩ {𝐵})) = ∅ ↔ (dom 𝐹 ∩ (𝐴 ∩ {𝐵})) = ∅)
8		elin 3927	. . . 4 ⊢ (𝐵 ∈ (dom 𝐹 ∩ 𝐴) ↔ (𝐵 ∈ dom 𝐹 ∧ 𝐵 ∈ 𝐴))
9		ancom 460	. . . 4 ⊢ ((𝐵 ∈ dom 𝐹 ∧ 𝐵 ∈ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 ∈ dom 𝐹))
10		snssi 4768	. . . . . . . . 9 ⊢ (𝐵 ∈ 𝐴 → {𝐵} ⊆ 𝐴)
11		incom 4168	. . . . . . . . . 10 ⊢ (𝐴 ∩ {𝐵}) = ({𝐵} ∩ 𝐴)
12		dfss2 3929	. . . . . . . . . . 11 ⊢ ({𝐵} ⊆ 𝐴 ↔ ({𝐵} ∩ 𝐴) = {𝐵})
13	12	biimpi 216	. . . . . . . . . 10 ⊢ ({𝐵} ⊆ 𝐴 → ({𝐵} ∩ 𝐴) = {𝐵})
14	11, 13	eqtrid 2776	. . . . . . . . 9 ⊢ ({𝐵} ⊆ 𝐴 → (𝐴 ∩ {𝐵}) = {𝐵})
15	10, 14	syl 17	. . . . . . . 8 ⊢ (𝐵 ∈ 𝐴 → (𝐴 ∩ {𝐵}) = {𝐵})
16	15	ineq2d 4179	. . . . . . 7 ⊢ (𝐵 ∈ 𝐴 → (dom 𝐹 ∩ (𝐴 ∩ {𝐵})) = (dom 𝐹 ∩ {𝐵}))
17	16	eqeq1d 2731	. . . . . 6 ⊢ (𝐵 ∈ 𝐴 → ((dom 𝐹 ∩ (𝐴 ∩ {𝐵})) = ∅ ↔ (dom 𝐹 ∩ {𝐵}) = ∅))
18		disjsn 4671	. . . . . 6 ⊢ ((dom 𝐹 ∩ {𝐵}) = ∅ ↔ ¬ 𝐵 ∈ dom 𝐹)
19	17, 18	bitrdi 287	. . . . 5 ⊢ (𝐵 ∈ 𝐴 → ((dom 𝐹 ∩ (𝐴 ∩ {𝐵})) = ∅ ↔ ¬ 𝐵 ∈ dom 𝐹))
20		disjsn 4671	. . . . . . . 8 ⊢ ((𝐴 ∩ {𝐵}) = ∅ ↔ ¬ 𝐵 ∈ 𝐴)
21	20	biimpri 228	. . . . . . 7 ⊢ (¬ 𝐵 ∈ 𝐴 → (𝐴 ∩ {𝐵}) = ∅)
22	21	ineq2d 4179	. . . . . 6 ⊢ (¬ 𝐵 ∈ 𝐴 → (dom 𝐹 ∩ (𝐴 ∩ {𝐵})) = (dom 𝐹 ∩ ∅))
23		in0 4354	. . . . . 6 ⊢ (dom 𝐹 ∩ ∅) = ∅
24	22, 23	eqtrdi 2780	. . . . 5 ⊢ (¬ 𝐵 ∈ 𝐴 → (dom 𝐹 ∩ (𝐴 ∩ {𝐵})) = ∅)
25	19, 24	resinsnlem 48832	. . . 4 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐵 ∈ dom 𝐹) ↔ ¬ (dom 𝐹 ∩ (𝐴 ∩ {𝐵})) = ∅)
26	8, 9, 25	3bitri 297	. . 3 ⊢ (𝐵 ∈ (dom 𝐹 ∩ 𝐴) ↔ ¬ (dom 𝐹 ∩ (𝐴 ∩ {𝐵})) = ∅)
27	26	con2bii 357	. 2 ⊢ ((dom 𝐹 ∩ (𝐴 ∩ {𝐵})) = ∅ ↔ ¬ 𝐵 ∈ (dom 𝐹 ∩ 𝐴))
28	3, 7, 27	3bitri 297	1 ⊢ ((𝐹 ↾ (𝐴 ∩ {𝐵})) = ∅ ↔ ¬ 𝐵 ∈ (dom 𝐹 ∩ 𝐴))

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ∩ cin 3910   ⊆ wss 3911  ∅c0 4292  {csn 4585  dom cdm 5631   ↾ cres 5633  Rel wrel 5636

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 5246  ax-nul 5256  ax-pr 5382

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 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-br 5103  df-opab 5165  df-xp 5637  df-rel 5638  df-dm 5641  df-res 5643

This theorem is referenced by: (None)