Theorem resinsnALT 50080

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 5995	. . 3 ⊢ Rel (𝐹 ↾ (𝐴 ∩ {𝐵}))
2		reldm0 5909	. . 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 6002	. . . 4 ⊢ dom (𝐹 ↾ (𝐴 ∩ {𝐵})) = ((𝐴 ∩ {𝐵}) ∩ dom 𝐹)
5		incom 4106	. . . 4 ⊢ ((𝐴 ∩ {𝐵}) ∩ dom 𝐹) = (dom 𝐹 ∩ (𝐴 ∩ {𝐵}))
6	4, 5	eqtri 2783	. . 3 ⊢ dom (𝐹 ↾ (𝐴 ∩ {𝐵})) = (dom 𝐹 ∩ (𝐴 ∩ {𝐵}))
7	6	eqeq1i 2765	. 2 ⊢ (dom (𝐹 ↾ (𝐴 ∩ {𝐵})) = ∅ ↔ (dom 𝐹 ∩ (𝐴 ∩ {𝐵})) = ∅)
8		elin 4093	. . . 4 ⊢ (𝐵 ∈ (dom 𝐹 ∩ 𝐴) ↔ (𝐵 ∈ dom 𝐹 ∧ 𝐵 ∈ 𝐴))
9		ancom 466	. . . 4 ⊢ ((𝐵 ∈ dom 𝐹 ∧ 𝐵 ∈ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 ∈ dom 𝐹))
10		snssi 4742	. . . . . . . . 9 ⊢ (𝐵 ∈ 𝐴 → {𝐵} ⊆ 𝐴)
11		incom 4106	. . . . . . . . . 10 ⊢ (𝐴 ∩ {𝐵}) = ({𝐵} ∩ 𝐴)
12		dfss2 4094	. . . . . . . . . . 11 ⊢ ({𝐵} ⊆ 𝐴 ↔ ({𝐵} ∩ 𝐴) = {𝐵})
13	12	biimpi 219	. . . . . . . . . 10 ⊢ ({𝐵} ⊆ 𝐴 → ({𝐵} ∩ 𝐴) = {𝐵})
14	11, 13	eqtrid 2807	. . . . . . . . 9 ⊢ ({𝐵} ⊆ 𝐴 → (𝐴 ∩ {𝐵}) = {𝐵})
15	10, 14	syl 18	. . . . . . . 8 ⊢ (𝐵 ∈ 𝐴 → (𝐴 ∩ {𝐵}) = {𝐵})
16	15	ineq2d 4117	. . . . . . 7 ⊢ (𝐵 ∈ 𝐴 → (dom 𝐹 ∩ (𝐴 ∩ {𝐵})) = (dom 𝐹 ∩ {𝐵}))
17	16	eqeq1d 2762	. . . . . 6 ⊢ (𝐵 ∈ 𝐴 → ((dom 𝐹 ∩ (𝐴 ∩ {𝐵})) = ∅ ↔ (dom 𝐹 ∩ {𝐵}) = ∅))
18		disjsn 4668	. . . . . 6 ⊢ ((dom 𝐹 ∩ {𝐵}) = ∅ ↔ ¬ 𝐵 ∈ dom 𝐹)
19	17, 18	bitrdi 290	. . . . 5 ⊢ (𝐵 ∈ 𝐴 → ((dom 𝐹 ∩ (𝐴 ∩ {𝐵})) = ∅ ↔ ¬ 𝐵 ∈ dom 𝐹))
20		disjsn 4668	. . . . . . . 8 ⊢ ((𝐴 ∩ {𝐵}) = ∅ ↔ ¬ 𝐵 ∈ 𝐴)
21	20	biimpri 231	. . . . . . 7 ⊢ (¬ 𝐵 ∈ 𝐴 → (𝐴 ∩ {𝐵}) = ∅)
22	21	ineq2d 4117	. . . . . 6 ⊢ (¬ 𝐵 ∈ 𝐴 → (dom 𝐹 ∩ (𝐴 ∩ {𝐵})) = (dom 𝐹 ∩ ∅))
23		in0 4344	. . . . . 6 ⊢ (dom 𝐹 ∩ ∅) = ∅
24	22, 23	eqtrdi 2811	. . . . 5 ⊢ (¬ 𝐵 ∈ 𝐴 → (dom 𝐹 ∩ (𝐴 ∩ {𝐵})) = ∅)
25	19, 24	resinsnlem 50078	. . . 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 401   = wceq 1569   ∈ wcel 2144   ⊆ wss 3895   ∩ cin 4089  ∅c0 4278  {csn 4584  dom cdm 5650   ↾ cres 5652  Rel wrel 5655

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-br 5103  df-opab 5167  df-xp 5656  df-rel 5657  df-dm 5660  df-res 5662

This theorem is used by: (None)