Theorem fnssintima 33676

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 4895	. . 3 ⊢ (𝐶 ⊆ ∩ (𝐹 “ 𝐵) ↔ ∀𝑦 ∈ (𝐹 “ 𝐵)𝐶 ⊆ 𝑦)
2		df-ral 3069	. . 3 ⊢ (∀𝑦 ∈ (𝐹 “ 𝐵)𝐶 ⊆ 𝑦 ↔ ∀𝑦(𝑦 ∈ (𝐹 “ 𝐵) → 𝐶 ⊆ 𝑦))
3	1, 2	bitri 274	. 2 ⊢ (𝐶 ⊆ ∩ (𝐹 “ 𝐵) ↔ ∀𝑦(𝑦 ∈ (𝐹 “ 𝐵) → 𝐶 ⊆ 𝑦))
4		fvelimab 6841	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝑦 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝑦))
5	4	imbi1d 342	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → ((𝑦 ∈ (𝐹 “ 𝐵) → 𝐶 ⊆ 𝑦) ↔ (∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝑦 → 𝐶 ⊆ 𝑦)))
6	5	albidv 1923	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∀𝑦(𝑦 ∈ (𝐹 “ 𝐵) → 𝐶 ⊆ 𝑦) ↔ ∀𝑦(∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝑦 → 𝐶 ⊆ 𝑦)))
7		ralcom4 3164	. . . 4 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦((𝐹‘𝑥) = 𝑦 → 𝐶 ⊆ 𝑦) ↔ ∀𝑦∀𝑥 ∈ 𝐵 ((𝐹‘𝑥) = 𝑦 → 𝐶 ⊆ 𝑦))
8		eqcom 2745	. . . . . . . 8 ⊢ ((𝐹‘𝑥) = 𝑦 ↔ 𝑦 = (𝐹‘𝑥))
9	8	imbi1i 350	. . . . . . 7 ⊢ (((𝐹‘𝑥) = 𝑦 → 𝐶 ⊆ 𝑦) ↔ (𝑦 = (𝐹‘𝑥) → 𝐶 ⊆ 𝑦))
10	9	albii 1822	. . . . . 6 ⊢ (∀𝑦((𝐹‘𝑥) = 𝑦 → 𝐶 ⊆ 𝑦) ↔ ∀𝑦(𝑦 = (𝐹‘𝑥) → 𝐶 ⊆ 𝑦))
11		fvex 6787	. . . . . . 7 ⊢ (𝐹‘𝑥) ∈ V
12		sseq2 3947	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝑥) → (𝐶 ⊆ 𝑦 ↔ 𝐶 ⊆ (𝐹‘𝑥)))
13	11, 12	ceqsalv 3467	. . . . . 6 ⊢ (∀𝑦(𝑦 = (𝐹‘𝑥) → 𝐶 ⊆ 𝑦) ↔ 𝐶 ⊆ (𝐹‘𝑥))
14	10, 13	bitri 274	. . . . 5 ⊢ (∀𝑦((𝐹‘𝑥) = 𝑦 → 𝐶 ⊆ 𝑦) ↔ 𝐶 ⊆ (𝐹‘𝑥))
15	14	ralbii 3092	. . . 4 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦((𝐹‘𝑥) = 𝑦 → 𝐶 ⊆ 𝑦) ↔ ∀𝑥 ∈ 𝐵 𝐶 ⊆ (𝐹‘𝑥))
16		r19.23v 3208	. . . . 5 ⊢ (∀𝑥 ∈ 𝐵 ((𝐹‘𝑥) = 𝑦 → 𝐶 ⊆ 𝑦) ↔ (∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝑦 → 𝐶 ⊆ 𝑦))
17	16	albii 1822	. . . 4 ⊢ (∀𝑦∀𝑥 ∈ 𝐵 ((𝐹‘𝑥) = 𝑦 → 𝐶 ⊆ 𝑦) ↔ ∀𝑦(∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝑦 → 𝐶 ⊆ 𝑦))
18	7, 15, 17	3bitr3ri 302	. . 3 ⊢ (∀𝑦(∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝑦 → 𝐶 ⊆ 𝑦) ↔ ∀𝑥 ∈ 𝐵 𝐶 ⊆ (𝐹‘𝑥))
19	6, 18	bitrdi 287	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∀𝑦(𝑦 ∈ (𝐹 “ 𝐵) → 𝐶 ⊆ 𝑦) ↔ ∀𝑥 ∈ 𝐵 𝐶 ⊆ (𝐹‘𝑥)))
20	3, 19	syl5bb 283	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐶 ⊆ ∩ (𝐹 “ 𝐵) ↔ ∀𝑥 ∈ 𝐵 𝐶 ⊆ (𝐹‘𝑥)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1537   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065   ⊆ wss 3887  ∩ cint 4879   “ cima 5592   Fn wfn 6428  ‘cfv 6433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-int 4880  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-fv 6441

This theorem is referenced by:  bday1s  34025  madebdaylemlrcut  34079