Theorem elintfv 33644

Description: Membership in an intersection of function values. (Contributed by Scott Fenton, 9-Dec-2021.)

Hypothesis

Ref	Expression
elintfv.1	⊢ 𝑋 ∈ V

Assertion

Ref	Expression
elintfv	⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝑋 ∈ ∩ (𝐹 “ 𝐵) ↔ ∀𝑦 ∈ 𝐵 𝑋 ∈ (𝐹‘𝑦)))

Distinct variable groups:   𝑦,𝐴   𝑦,𝐵   𝑦,𝐹   𝑦,𝑋

Proof of Theorem elintfv

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elintfv.1	. . 3 ⊢ 𝑋 ∈ V
2	1	elint 4882	. 2 ⊢ (𝑋 ∈ ∩ (𝐹 “ 𝐵) ↔ ∀𝑧(𝑧 ∈ (𝐹 “ 𝐵) → 𝑋 ∈ 𝑧))
3		fvelimab 6823	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝑧 ∈ (𝐹 “ 𝐵) ↔ ∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑧))
4	3	imbi1d 341	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → ((𝑧 ∈ (𝐹 “ 𝐵) → 𝑋 ∈ 𝑧) ↔ (∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑧 → 𝑋 ∈ 𝑧)))
5		r19.23v 3207	. . . . 5 ⊢ (∀𝑦 ∈ 𝐵 ((𝐹‘𝑦) = 𝑧 → 𝑋 ∈ 𝑧) ↔ (∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑧 → 𝑋 ∈ 𝑧))
6	4, 5	bitr4di 288	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → ((𝑧 ∈ (𝐹 “ 𝐵) → 𝑋 ∈ 𝑧) ↔ ∀𝑦 ∈ 𝐵 ((𝐹‘𝑦) = 𝑧 → 𝑋 ∈ 𝑧)))
7	6	albidv 1924	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∀𝑧(𝑧 ∈ (𝐹 “ 𝐵) → 𝑋 ∈ 𝑧) ↔ ∀𝑧∀𝑦 ∈ 𝐵 ((𝐹‘𝑦) = 𝑧 → 𝑋 ∈ 𝑧)))
8		ralcom4 3161	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧((𝐹‘𝑦) = 𝑧 → 𝑋 ∈ 𝑧) ↔ ∀𝑧∀𝑦 ∈ 𝐵 ((𝐹‘𝑦) = 𝑧 → 𝑋 ∈ 𝑧))
9		eqcom 2745	. . . . . . . 8 ⊢ ((𝐹‘𝑦) = 𝑧 ↔ 𝑧 = (𝐹‘𝑦))
10	9	imbi1i 349	. . . . . . 7 ⊢ (((𝐹‘𝑦) = 𝑧 → 𝑋 ∈ 𝑧) ↔ (𝑧 = (𝐹‘𝑦) → 𝑋 ∈ 𝑧))
11	10	albii 1823	. . . . . 6 ⊢ (∀𝑧((𝐹‘𝑦) = 𝑧 → 𝑋 ∈ 𝑧) ↔ ∀𝑧(𝑧 = (𝐹‘𝑦) → 𝑋 ∈ 𝑧))
12		fvex 6769	. . . . . . 7 ⊢ (𝐹‘𝑦) ∈ V
13		eleq2 2827	. . . . . . 7 ⊢ (𝑧 = (𝐹‘𝑦) → (𝑋 ∈ 𝑧 ↔ 𝑋 ∈ (𝐹‘𝑦)))
14	12, 13	ceqsalv 3457	. . . . . 6 ⊢ (∀𝑧(𝑧 = (𝐹‘𝑦) → 𝑋 ∈ 𝑧) ↔ 𝑋 ∈ (𝐹‘𝑦))
15	11, 14	bitri 274	. . . . 5 ⊢ (∀𝑧((𝐹‘𝑦) = 𝑧 → 𝑋 ∈ 𝑧) ↔ 𝑋 ∈ (𝐹‘𝑦))
16	15	ralbii 3090	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧((𝐹‘𝑦) = 𝑧 → 𝑋 ∈ 𝑧) ↔ ∀𝑦 ∈ 𝐵 𝑋 ∈ (𝐹‘𝑦))
17	8, 16	bitr3i 276	. . 3 ⊢ (∀𝑧∀𝑦 ∈ 𝐵 ((𝐹‘𝑦) = 𝑧 → 𝑋 ∈ 𝑧) ↔ ∀𝑦 ∈ 𝐵 𝑋 ∈ (𝐹‘𝑦))
18	7, 17	bitrdi 286	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∀𝑧(𝑧 ∈ (𝐹 “ 𝐵) → 𝑋 ∈ 𝑧) ↔ ∀𝑦 ∈ 𝐵 𝑋 ∈ (𝐹‘𝑦)))
19	2, 18	syl5bb 282	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝑋 ∈ ∩ (𝐹 “ 𝐵) ↔ ∀𝑦 ∈ 𝐵 𝑋 ∈ (𝐹‘𝑦)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395  ∀wal 1537   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064  Vcvv 3422   ⊆ wss 3883  ∩ cint 4876   “ cima 5583   Fn wfn 6413  ‘cfv 6418

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-int 4877  df-br 5071  df-opab 5133  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-fv 6426

This theorem is referenced by: (None)