Theorem resinsnALT 49118

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 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		dmres 5971	. . . 4 ⊢ dom (𝐹 ↾ (𝐴 ∩ {𝐵})) = ((𝐴 ∩ {𝐵}) ∩ dom 𝐹)
5		incom 4161	. . . 4 ⊢ ((𝐴 ∩ {𝐵}) ∩ dom 𝐹) = (dom 𝐹 ∩ (𝐴 ∩ {𝐵}))
6	4, 5	eqtri 2759	. . 3 ⊢ dom (𝐹 ↾ (𝐴 ∩ {𝐵})) = (dom 𝐹 ∩ (𝐴 ∩ {𝐵}))
7	6	eqeq1i 2741	. 2 ⊢ (dom (𝐹 ↾ (𝐴 ∩ {𝐵})) = ∅ ↔ (dom 𝐹 ∩ (𝐴 ∩ {𝐵})) = ∅)
8		elin 3917	. . . 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 4764	. . . . . . . . 9 ⊢ (𝐵 ∈ 𝐴 → {𝐵} ⊆ 𝐴)
11		incom 4161	. . . . . . . . . 10 ⊢ (𝐴 ∩ {𝐵}) = ({𝐵} ∩ 𝐴)
12		dfss2 3919	. . . . . . . . . . 11 ⊢ ({𝐵} ⊆ 𝐴 ↔ ({𝐵} ∩ 𝐴) = {𝐵})
13	12	biimpi 216	. . . . . . . . . 10 ⊢ ({𝐵} ⊆ 𝐴 → ({𝐵} ∩ 𝐴) = {𝐵})
14	11, 13	eqtrid 2783	. . . . . . . . 9 ⊢ ({𝐵} ⊆ 𝐴 → (𝐴 ∩ {𝐵}) = {𝐵})
15	10, 14	syl 17	. . . . . . . 8 ⊢ (𝐵 ∈ 𝐴 → (𝐴 ∩ {𝐵}) = {𝐵})
16	15	ineq2d 4172	. . . . . . 7 ⊢ (𝐵 ∈ 𝐴 → (dom 𝐹 ∩ (𝐴 ∩ {𝐵})) = (dom 𝐹 ∩ {𝐵}))
17	16	eqeq1d 2738	. . . . . 6 ⊢ (𝐵 ∈ 𝐴 → ((dom 𝐹 ∩ (𝐴 ∩ {𝐵})) = ∅ ↔ (dom 𝐹 ∩ {𝐵}) = ∅))
18		disjsn 4668	. . . . . 6 ⊢ ((dom 𝐹 ∩ {𝐵}) = ∅ ↔ ¬ 𝐵 ∈ dom 𝐹)
19	17, 18	bitrdi 287	. . . . 5 ⊢ (𝐵 ∈ 𝐴 → ((dom 𝐹 ∩ (𝐴 ∩ {𝐵})) = ∅ ↔ ¬ 𝐵 ∈ dom 𝐹))
20		disjsn 4668	. . . . . . . 8 ⊢ ((𝐴 ∩ {𝐵}) = ∅ ↔ ¬ 𝐵 ∈ 𝐴)
21	20	biimpri 228	. . . . . . 7 ⊢ (¬ 𝐵 ∈ 𝐴 → (𝐴 ∩ {𝐵}) = ∅)
22	21	ineq2d 4172	. . . . . 6 ⊢ (¬ 𝐵 ∈ 𝐴 → (dom 𝐹 ∩ (𝐴 ∩ {𝐵})) = (dom 𝐹 ∩ ∅))
23		in0 4347	. . . . . 6 ⊢ (dom 𝐹 ∩ ∅) = ∅
24	22, 23	eqtrdi 2787	. . . . 5 ⊢ (¬ 𝐵 ∈ 𝐴 → (dom 𝐹 ∩ (𝐴 ∩ {𝐵})) = ∅)
25	19, 24	resinsnlem 49116	. . . 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 1541   ∈ wcel 2113   ∩ cin 3900   ⊆ wss 3901  ∅c0 4285  {csn 4580  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 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-xp 5630  df-rel 5631  df-dm 5634  df-res 5636

This theorem is referenced by: (None)