Theorem elintfv 36530

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 4913	. 2 ⊢ (𝑋 ∈ ∩ (𝐹 “ 𝐵) ↔ ∀𝑧(𝑧 ∈ (𝐹 “ 𝐵) → 𝑋 ∈ 𝑧))
3		fvelimab 6957	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝑧 ∈ (𝐹 “ 𝐵) ↔ ∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑧))
4	3	imbi1d 344	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → ((𝑧 ∈ (𝐹 “ 𝐵) → 𝑋 ∈ 𝑧) ↔ (∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑧 → 𝑋 ∈ 𝑧)))
5		r19.23v 3190	. . . . 5 ⊢ (∀𝑦 ∈ 𝐵 ((𝐹‘𝑦) = 𝑧 → 𝑋 ∈ 𝑧) ↔ (∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑧 → 𝑋 ∈ 𝑧))
6	4, 5	bitr4di 292	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → ((𝑧 ∈ (𝐹 “ 𝐵) → 𝑋 ∈ 𝑧) ↔ ∀𝑦 ∈ 𝐵 ((𝐹‘𝑦) = 𝑧 → 𝑋 ∈ 𝑧)))
7	6	albidv 1953	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∀𝑧(𝑧 ∈ (𝐹 “ 𝐵) → 𝑋 ∈ 𝑧) ↔ ∀𝑧∀𝑦 ∈ 𝐵 ((𝐹‘𝑦) = 𝑧 → 𝑋 ∈ 𝑧)))
8		ralcom4 3289	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧((𝐹‘𝑦) = 𝑧 → 𝑋 ∈ 𝑧) ↔ ∀𝑧∀𝑦 ∈ 𝐵 ((𝐹‘𝑦) = 𝑧 → 𝑋 ∈ 𝑧))
9		eqcom 2768	. . . . . . . 8 ⊢ ((𝐹‘𝑦) = 𝑧 ↔ 𝑧 = (𝐹‘𝑦))
10	9	imbi1i 352	. . . . . . 7 ⊢ (((𝐹‘𝑦) = 𝑧 → 𝑋 ∈ 𝑧) ↔ (𝑧 = (𝐹‘𝑦) → 𝑋 ∈ 𝑧))
11	10	albii 1852	. . . . . 6 ⊢ (∀𝑧((𝐹‘𝑦) = 𝑧 → 𝑋 ∈ 𝑧) ↔ ∀𝑧(𝑧 = (𝐹‘𝑦) → 𝑋 ∈ 𝑧))
12		fvex 6898	. . . . . . 7 ⊢ (𝐹‘𝑦) ∈ V
13		eleq2 2850	. . . . . . 7 ⊢ (𝑧 = (𝐹‘𝑦) → (𝑋 ∈ 𝑧 ↔ 𝑋 ∈ (𝐹‘𝑦)))
14	12, 13	ceqsalv 3490	. . . . . 6 ⊢ (∀𝑧(𝑧 = (𝐹‘𝑦) → 𝑋 ∈ 𝑧) ↔ 𝑋 ∈ (𝐹‘𝑦))
15	11, 14	bitri 278	. . . . 5 ⊢ (∀𝑧((𝐹‘𝑦) = 𝑧 → 𝑋 ∈ 𝑧) ↔ 𝑋 ∈ (𝐹‘𝑦))
16	15	ralbii 3109	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧((𝐹‘𝑦) = 𝑧 → 𝑋 ∈ 𝑧) ↔ ∀𝑦 ∈ 𝐵 𝑋 ∈ (𝐹‘𝑦))
17	8, 16	bitr3i 280	. . 3 ⊢ (∀𝑧∀𝑦 ∈ 𝐵 ((𝐹‘𝑦) = 𝑧 → 𝑋 ∈ 𝑧) ↔ ∀𝑦 ∈ 𝐵 𝑋 ∈ (𝐹‘𝑦))
18	7, 17	bitrdi 290	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∀𝑧(𝑧 ∈ (𝐹 “ 𝐵) → 𝑋 ∈ 𝑧) ↔ ∀𝑦 ∈ 𝐵 𝑋 ∈ (𝐹‘𝑦)))
19	2, 18	bitrid 286	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝑋 ∈ ∩ (𝐹 “ 𝐵) ↔ ∀𝑦 ∈ 𝐵 𝑋 ∈ (𝐹‘𝑦)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401  ∀wal 1568   = wceq 1570   ∈ wcel 2145  ∀wral 3077  ∃wrex 3087  Vcvv 3451   ⊆ wss 3899  ∩ cint 4907   “ cima 5654   Fn wfn 6533  ‘cfv 6538

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 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-br 5104  df-opab 5168  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6494  df-fun 6540  df-fn 6541  df-fv 6546

This theorem is used by: (None)