Theorem fnssintima 7369

Description: Condition for subset of an intersection of an image. (Contributed by Scott Fenton, 16-Aug-2024.)

Assertion

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

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

Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem fnssintima

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssint 4927	. . 3 ⊢ (𝐶 ⊆ ∩ (𝐹 “ 𝐵) ↔ ∀𝑦 ∈ (𝐹 “ 𝐵)𝐶 ⊆ 𝑦)
2		df-ral 3079	. . 3 ⊢ (∀𝑦 ∈ (𝐹 “ 𝐵)𝐶 ⊆ 𝑦 ↔ ∀𝑦(𝑦 ∈ (𝐹 “ 𝐵) → 𝐶 ⊆ 𝑦))
3	1, 2	bitri 278	. 2 ⊢ (𝐶 ⊆ ∩ (𝐹 “ 𝐵) ↔ ∀𝑦(𝑦 ∈ (𝐹 “ 𝐵) → 𝐶 ⊆ 𝑦))
4		fvelimab 6954	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝑦 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝑦))
5	4	imbi1d 344	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → ((𝑦 ∈ (𝐹 “ 𝐵) → 𝐶 ⊆ 𝑦) ↔ (∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝑦 → 𝐶 ⊆ 𝑦)))
6	5	albidv 1953	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∀𝑦(𝑦 ∈ (𝐹 “ 𝐵) → 𝐶 ⊆ 𝑦) ↔ ∀𝑦(∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝑦 → 𝐶 ⊆ 𝑦)))
7		ralcom4 3290	. . . 4 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦((𝐹‘𝑥) = 𝑦 → 𝐶 ⊆ 𝑦) ↔ ∀𝑦∀𝑥 ∈ 𝐵 ((𝐹‘𝑥) = 𝑦 → 𝐶 ⊆ 𝑦))
8		eqcom 2769	. . . . . . . 8 ⊢ ((𝐹‘𝑥) = 𝑦 ↔ 𝑦 = (𝐹‘𝑥))
9	8	imbi1i 352	. . . . . . 7 ⊢ (((𝐹‘𝑥) = 𝑦 → 𝐶 ⊆ 𝑦) ↔ (𝑦 = (𝐹‘𝑥) → 𝐶 ⊆ 𝑦))
10	9	albii 1852	. . . . . 6 ⊢ (∀𝑦((𝐹‘𝑥) = 𝑦 → 𝐶 ⊆ 𝑦) ↔ ∀𝑦(𝑦 = (𝐹‘𝑥) → 𝐶 ⊆ 𝑦))
11		fvex 6895	. . . . . . 7 ⊢ (𝐹‘𝑥) ∈ V
12		sseq2 3960	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝑥) → (𝐶 ⊆ 𝑦 ↔ 𝐶 ⊆ (𝐹‘𝑥)))
13	11, 12	ceqsalv 3492	. . . . . 6 ⊢ (∀𝑦(𝑦 = (𝐹‘𝑥) → 𝐶 ⊆ 𝑦) ↔ 𝐶 ⊆ (𝐹‘𝑥))
14	10, 13	bitri 278	. . . . 5 ⊢ (∀𝑦((𝐹‘𝑥) = 𝑦 → 𝐶 ⊆ 𝑦) ↔ 𝐶 ⊆ (𝐹‘𝑥))
15	14	ralbii 3110	. . . 4 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦((𝐹‘𝑥) = 𝑦 → 𝐶 ⊆ 𝑦) ↔ ∀𝑥 ∈ 𝐵 𝐶 ⊆ (𝐹‘𝑥))
16		r19.23v 3191	. . . . 5 ⊢ (∀𝑥 ∈ 𝐵 ((𝐹‘𝑥) = 𝑦 → 𝐶 ⊆ 𝑦) ↔ (∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝑦 → 𝐶 ⊆ 𝑦))
17	16	albii 1852	. . . 4 ⊢ (∀𝑦∀𝑥 ∈ 𝐵 ((𝐹‘𝑥) = 𝑦 → 𝐶 ⊆ 𝑦) ↔ ∀𝑦(∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝑦 → 𝐶 ⊆ 𝑦))
18	7, 15, 17	3bitr3ri 305	. . 3 ⊢ (∀𝑦(∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝑦 → 𝐶 ⊆ 𝑦) ↔ ∀𝑥 ∈ 𝐵 𝐶 ⊆ (𝐹‘𝑥))
19	6, 18	bitrdi 290	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∀𝑦(𝑦 ∈ (𝐹 “ 𝐵) → 𝐶 ⊆ 𝑦) ↔ ∀𝑥 ∈ 𝐵 𝐶 ⊆ (𝐹‘𝑥)))
20	3, 19	bitrid 286	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐶 ⊆ ∩ (𝐹 “ 𝐵) ↔ ∀𝑥 ∈ 𝐵 𝐶 ⊆ (𝐹‘𝑥)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570   ∈ wcel 2145  ∀wral 3078  ∃wrex 3088   ⊆ 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:  bday1  28087  madebdaylemlrcut  28172  fnfvintima  35599