Theorem fnssintima 7361

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 4968	. . 3 ⊢ (𝐶 ⊆ ∩ (𝐹 “ 𝐵) ↔ ∀𝑦 ∈ (𝐹 “ 𝐵)𝐶 ⊆ 𝑦)
2		df-ral 3062	. . 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 6964	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝑦 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝑦))
5	4	imbi1d 341	. . . 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 3283	. . . 4 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦((𝐹‘𝑥) = 𝑦 → 𝐶 ⊆ 𝑦) ↔ ∀𝑦∀𝑥 ∈ 𝐵 ((𝐹‘𝑥) = 𝑦 → 𝐶 ⊆ 𝑦))
8		eqcom 2739	. . . . . . . 8 ⊢ ((𝐹‘𝑥) = 𝑦 ↔ 𝑦 = (𝐹‘𝑥))
9	8	imbi1i 349	. . . . . . 7 ⊢ (((𝐹‘𝑥) = 𝑦 → 𝐶 ⊆ 𝑦) ↔ (𝑦 = (𝐹‘𝑥) → 𝐶 ⊆ 𝑦))
10	9	albii 1821	. . . . . 6 ⊢ (∀𝑦((𝐹‘𝑥) = 𝑦 → 𝐶 ⊆ 𝑦) ↔ ∀𝑦(𝑦 = (𝐹‘𝑥) → 𝐶 ⊆ 𝑦))
11		fvex 6904	. . . . . . 7 ⊢ (𝐹‘𝑥) ∈ V
12		sseq2 4008	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝑥) → (𝐶 ⊆ 𝑦 ↔ 𝐶 ⊆ (𝐹‘𝑥)))
13	11, 12	ceqsalv 3511	. . . . . 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 3093	. . . 4 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦((𝐹‘𝑥) = 𝑦 → 𝐶 ⊆ 𝑦) ↔ ∀𝑥 ∈ 𝐵 𝐶 ⊆ (𝐹‘𝑥))
16		r19.23v 3182	. . . . 5 ⊢ (∀𝑥 ∈ 𝐵 ((𝐹‘𝑥) = 𝑦 → 𝐶 ⊆ 𝑦) ↔ (∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝑦 → 𝐶 ⊆ 𝑦))
17	16	albii 1821	. . . 4 ⊢ (∀𝑦∀𝑥 ∈ 𝐵 ((𝐹‘𝑥) = 𝑦 → 𝐶 ⊆ 𝑦) ↔ ∀𝑦(∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝑦 → 𝐶 ⊆ 𝑦))
18	7, 15, 17	3bitr3ri 301	. . 3 ⊢ (∀𝑦(∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝑦 → 𝐶 ⊆ 𝑦) ↔ ∀𝑥 ∈ 𝐵 𝐶 ⊆ (𝐹‘𝑥))
19	6, 18	bitrdi 286	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∀𝑦(𝑦 ∈ (𝐹 “ 𝐵) → 𝐶 ⊆ 𝑦) ↔ ∀𝑥 ∈ 𝐵 𝐶 ⊆ (𝐹‘𝑥)))
20	3, 19	bitrid 282	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐶 ⊆ ∩ (𝐹 “ 𝐵) ↔ ∀𝑥 ∈ 𝐵 𝐶 ⊆ (𝐹‘𝑥)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396  ∀wal 1539   = wceq 1541   ∈ wcel 2106  ∀wral 3061  ∃wrex 3070   ⊆ wss 3948  ∩ cint 4950   “ cima 5679   Fn wfn 6538  ‘cfv 6543

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-fv 6551

This theorem is referenced by:  bday1s  27340  madebdaylemlrcut  27401