Theorem fnfvintima 35578

Description: Condition for a function value to equal the intersection of an image that contains it. (Contributed by BTernaryTau, 23-Jun-2026.)

Assertion

Ref	Expression
fnfvintima	⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ∈ 𝐵) → ((𝐹‘𝐶) = ∩ (𝐹 “ 𝐵) ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝐶) ⊆ (𝐹‘𝑥)))

Distinct variable groups:   𝑥,𝐵   𝑥,𝐶   𝑥,𝐹

Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem fnfvintima

Step	Hyp	Ref	Expression
1		eqimss 3992	. . 3 ⊢ ((𝐹‘𝐶) = ∩ (𝐹 “ 𝐵) → (𝐹‘𝐶) ⊆ ∩ (𝐹 “ 𝐵))
2		fnssintima 7368	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → ((𝐹‘𝐶) ⊆ ∩ (𝐹 “ 𝐵) ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝐶) ⊆ (𝐹‘𝑥)))
3	2	3adant3 1150	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ∈ 𝐵) → ((𝐹‘𝐶) ⊆ ∩ (𝐹 “ 𝐵) ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝐶) ⊆ (𝐹‘𝑥)))
4	1, 3	imbitrid 247	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ∈ 𝐵) → ((𝐹‘𝐶) = ∩ (𝐹 “ 𝐵) → ∀𝑥 ∈ 𝐵 (𝐹‘𝐶) ⊆ (𝐹‘𝑥)))
5	3	biimprd 251	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ∈ 𝐵) → (∀𝑥 ∈ 𝐵 (𝐹‘𝐶) ⊆ (𝐹‘𝑥) → (𝐹‘𝐶) ⊆ ∩ (𝐹 “ 𝐵)))
6		fnfvima 7235	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ∈ 𝐵) → (𝐹‘𝐶) ∈ (𝐹 “ 𝐵))
7		intss1 4926	. . . . 5 ⊢ ((𝐹‘𝐶) ∈ (𝐹 “ 𝐵) → ∩ (𝐹 “ 𝐵) ⊆ (𝐹‘𝐶))
8	6, 7	syl 18	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ∈ 𝐵) → ∩ (𝐹 “ 𝐵) ⊆ (𝐹‘𝐶))
9	5, 8	jctird 536	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ∈ 𝐵) → (∀𝑥 ∈ 𝐵 (𝐹‘𝐶) ⊆ (𝐹‘𝑥) → ((𝐹‘𝐶) ⊆ ∩ (𝐹 “ 𝐵) ∧ ∩ (𝐹 “ 𝐵) ⊆ (𝐹‘𝐶))))
10		eqss 3949	. . 3 ⊢ ((𝐹‘𝐶) = ∩ (𝐹 “ 𝐵) ↔ ((𝐹‘𝐶) ⊆ ∩ (𝐹 “ 𝐵) ∧ ∩ (𝐹 “ 𝐵) ⊆ (𝐹‘𝐶)))
11	9, 10	imbitrrdi 255	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ∈ 𝐵) → (∀𝑥 ∈ 𝐵 (𝐹‘𝐶) ⊆ (𝐹‘𝑥) → (𝐹‘𝐶) = ∩ (𝐹 “ 𝐵)))
12	4, 11	impbid 215	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴 ∧ 𝐶 ∈ 𝐵) → ((𝐹‘𝐶) = ∩ (𝐹 “ 𝐵) ↔ ∀𝑥 ∈ 𝐵 (𝐹‘𝐶) ⊆ (𝐹‘𝑥)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3078   ⊆ wss 3902  ∩ cint 4910   “ cima 5662   Fn wfn 6532  ‘cfv 6537

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-int 4911  df-br 5108  df-opab 5172  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-fv 6545

This theorem is used by:  dfscott2  35612