Theorem resinsnALT 49679

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 6003	. . 3 ⊢ Rel (𝐹 ↾ (𝐴 ∩ {𝐵}))
2		reldm0 5917	. . 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 6010	. . . 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 2785	. . 3 ⊢ dom (𝐹 ↾ (𝐴 ∩ {𝐵})) = (dom 𝐹 ∩ (𝐴 ∩ {𝐵}))
7	6	eqeq1i 2767	. 2 ⊢ (dom (𝐹 ↾ (𝐴 ∩ {𝐵})) = ∅ ↔ (dom 𝐹 ∩ (𝐴 ∩ {𝐵})) = ∅)
8		elin 3920	. . . 4 ⊢ (𝐵 ∈ (dom 𝐹 ∩ 𝐴) ↔ (𝐵 ∈ dom 𝐹 ∧ 𝐵 ∈ 𝐴))
9		ancom 465	. . . 4 ⊢ ((𝐵 ∈ dom 𝐹 ∧ 𝐵 ∈ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 ∈ dom 𝐹))
10		snssi 4750	. . . . . . . . 9 ⊢ (𝐵 ∈ 𝐴 → {𝐵} ⊆ 𝐴)
11		incom 4161	. . . . . . . . . 10 ⊢ (𝐴 ∩ {𝐵}) = ({𝐵} ∩ 𝐴)
12		dfss2 3922	. . . . . . . . . . 11 ⊢ ({𝐵} ⊆ 𝐴 ↔ ({𝐵} ∩ 𝐴) = {𝐵})
13	12	biimpi 219	. . . . . . . . . 10 ⊢ ({𝐵} ⊆ 𝐴 → ({𝐵} ∩ 𝐴) = {𝐵})
14	11, 13	eqtrid 2809	. . . . . . . . 9 ⊢ ({𝐵} ⊆ 𝐴 → (𝐴 ∩ {𝐵}) = {𝐵})
15	10, 14	syl 18	. . . . . . . 8 ⊢ (𝐵 ∈ 𝐴 → (𝐴 ∩ {𝐵}) = {𝐵})
16	15	ineq2d 4172	. . . . . . 7 ⊢ (𝐵 ∈ 𝐴 → (dom 𝐹 ∩ (𝐴 ∩ {𝐵})) = (dom 𝐹 ∩ {𝐵}))
17	16	eqeq1d 2764	. . . . . 6 ⊢ (𝐵 ∈ 𝐴 → ((dom 𝐹 ∩ (𝐴 ∩ {𝐵})) = ∅ ↔ (dom 𝐹 ∩ {𝐵}) = ∅))
18		disjsn 4676	. . . . . 6 ⊢ ((dom 𝐹 ∩ {𝐵}) = ∅ ↔ ¬ 𝐵 ∈ dom 𝐹)
19	17, 18	bitrdi 290	. . . . 5 ⊢ (𝐵 ∈ 𝐴 → ((dom 𝐹 ∩ (𝐴 ∩ {𝐵})) = ∅ ↔ ¬ 𝐵 ∈ dom 𝐹))
20		disjsn 4676	. . . . . . . 8 ⊢ ((𝐴 ∩ {𝐵}) = ∅ ↔ ¬ 𝐵 ∈ 𝐴)
21	20	biimpri 231	. . . . . . 7 ⊢ (¬ 𝐵 ∈ 𝐴 → (𝐴 ∩ {𝐵}) = ∅)
22	21	ineq2d 4172	. . . . . 6 ⊢ (¬ 𝐵 ∈ 𝐴 → (dom 𝐹 ∩ (𝐴 ∩ {𝐵})) = (dom 𝐹 ∩ ∅))
23		in0 4351	. . . . . 6 ⊢ (dom 𝐹 ∩ ∅) = ∅
24	22, 23	eqtrdi 2813	. . . . 5 ⊢ (¬ 𝐵 ∈ 𝐴 → (dom 𝐹 ∩ (𝐴 ∩ {𝐵})) = ∅)
25	19, 24	resinsnlem 49677	. . . 4 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐵 ∈ dom 𝐹) ↔ ¬ (dom 𝐹 ∩ (𝐴 ∩ {𝐵})) = ∅)
26	8, 9, 25	3bitri 300	. . 3 ⊢ (𝐵 ∈ (dom 𝐹 ∩ 𝐴) ↔ ¬ (dom 𝐹 ∩ (𝐴 ∩ {𝐵})) = ∅)
27	26	con2bii 360	. 2 ⊢ ((dom 𝐹 ∩ (𝐴 ∩ {𝐵})) = ∅ ↔ ¬ 𝐵 ∈ (dom 𝐹 ∩ 𝐴))
28	3, 7, 27	3bitri 300	1 ⊢ ((𝐹 ↾ (𝐴 ∩ {𝐵})) = ∅ ↔ ¬ 𝐵 ∈ (dom 𝐹 ∩ 𝐴))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 400   = wceq 1569   ∈ wcel 2142   ∩ cin 3903   ⊆ wss 3904  ∅c0 4285  {csn 4588  dom cdm 5660   ↾ cres 5662  Rel wrel 5665

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5256  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-xp 5666  df-rel 5667  df-dm 5670  df-res 5672

This theorem is used by: (None)