Theorem fnssintima 7337

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 4928	. . 3 ⊢ (𝐶 ⊆ ∩ (𝐹 “ 𝐵) ↔ ∀𝑦 ∈ (𝐹 “ 𝐵)𝐶 ⊆ 𝑦)
2		df-ral 3045	. . 3 ⊢ (∀𝑦 ∈ (𝐹 “ 𝐵)𝐶 ⊆ 𝑦 ↔ ∀𝑦(𝑦 ∈ (𝐹 “ 𝐵) → 𝐶 ⊆ 𝑦))
3	1, 2	bitri 275	. 2 ⊢ (𝐶 ⊆ ∩ (𝐹 “ 𝐵) ↔ ∀𝑦(𝑦 ∈ (𝐹 “ 𝐵) → 𝐶 ⊆ 𝑦))
4		fvelimab 6933	. . . . 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 1920	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∀𝑦(𝑦 ∈ (𝐹 “ 𝐵) → 𝐶 ⊆ 𝑦) ↔ ∀𝑦(∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝑦 → 𝐶 ⊆ 𝑦)))
7		ralcom4 3263	. . . 4 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦((𝐹‘𝑥) = 𝑦 → 𝐶 ⊆ 𝑦) ↔ ∀𝑦∀𝑥 ∈ 𝐵 ((𝐹‘𝑥) = 𝑦 → 𝐶 ⊆ 𝑦))
8		eqcom 2736	. . . . . . . 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 1819	. . . . . 6 ⊢ (∀𝑦((𝐹‘𝑥) = 𝑦 → 𝐶 ⊆ 𝑦) ↔ ∀𝑦(𝑦 = (𝐹‘𝑥) → 𝐶 ⊆ 𝑦))
11		fvex 6871	. . . . . . 7 ⊢ (𝐹‘𝑥) ∈ V
12		sseq2 3973	. . . . . . 7 ⊢ (𝑦 = (𝐹‘𝑥) → (𝐶 ⊆ 𝑦 ↔ 𝐶 ⊆ (𝐹‘𝑥)))
13	11, 12	ceqsalv 3487	. . . . . 6 ⊢ (∀𝑦(𝑦 = (𝐹‘𝑥) → 𝐶 ⊆ 𝑦) ↔ 𝐶 ⊆ (𝐹‘𝑥))
14	10, 13	bitri 275	. . . . 5 ⊢ (∀𝑦((𝐹‘𝑥) = 𝑦 → 𝐶 ⊆ 𝑦) ↔ 𝐶 ⊆ (𝐹‘𝑥))
15	14	ralbii 3075	. . . 4 ⊢ (∀𝑥 ∈ 𝐵 ∀𝑦((𝐹‘𝑥) = 𝑦 → 𝐶 ⊆ 𝑦) ↔ ∀𝑥 ∈ 𝐵 𝐶 ⊆ (𝐹‘𝑥))
16		r19.23v 3161	. . . . 5 ⊢ (∀𝑥 ∈ 𝐵 ((𝐹‘𝑥) = 𝑦 → 𝐶 ⊆ 𝑦) ↔ (∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝑦 → 𝐶 ⊆ 𝑦))
17	16	albii 1819	. . . 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	bitrid 283	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝐶 ⊆ ∩ (𝐹 “ 𝐵) ↔ ∀𝑥 ∈ 𝐵 𝐶 ⊆ (𝐹‘𝑥)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395  ∀wal 1538   = wceq 1540   ∈ wcel 2109  ∀wral 3044  ∃wrex 3053   ⊆ wss 3914  ∩ cint 4910   “ cima 5641   Fn wfn 6506  ‘cfv 6511

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-int 4911  df-br 5108  df-opab 5170  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-fv 6519

This theorem is referenced by:  bday1s  27743  madebdaylemlrcut  27810