Theorem elintfv 36115

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 4911	. 2 ⊢ (𝑋 ∈ ∩ (𝐹 “ 𝐵) ↔ ∀𝑧(𝑧 ∈ (𝐹 “ 𝐵) → 𝑋 ∈ 𝑧))
3		fvelimab 6939	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝑧 ∈ (𝐹 “ 𝐵) ↔ ∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑧))
4	3	imbi1d 343	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → ((𝑧 ∈ (𝐹 “ 𝐵) → 𝑋 ∈ 𝑧) ↔ (∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑧 → 𝑋 ∈ 𝑧)))
5		r19.23v 3189	. . . . 5 ⊢ (∀𝑦 ∈ 𝐵 ((𝐹‘𝑦) = 𝑧 → 𝑋 ∈ 𝑧) ↔ (∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑧 → 𝑋 ∈ 𝑧))
6	4, 5	bitr4di 291	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → ((𝑧 ∈ (𝐹 “ 𝐵) → 𝑋 ∈ 𝑧) ↔ ∀𝑦 ∈ 𝐵 ((𝐹‘𝑦) = 𝑧 → 𝑋 ∈ 𝑧)))
7	6	albidv 1940	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∀𝑧(𝑧 ∈ (𝐹 “ 𝐵) → 𝑋 ∈ 𝑧) ↔ ∀𝑧∀𝑦 ∈ 𝐵 ((𝐹‘𝑦) = 𝑧 → 𝑋 ∈ 𝑧)))
8		ralcom4 3288	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧((𝐹‘𝑦) = 𝑧 → 𝑋 ∈ 𝑧) ↔ ∀𝑧∀𝑦 ∈ 𝐵 ((𝐹‘𝑦) = 𝑧 → 𝑋 ∈ 𝑧))
9		eqcom 2769	. . . . . . . 8 ⊢ ((𝐹‘𝑦) = 𝑧 ↔ 𝑧 = (𝐹‘𝑦))
10	9	imbi1i 351	. . . . . . 7 ⊢ (((𝐹‘𝑦) = 𝑧 → 𝑋 ∈ 𝑧) ↔ (𝑧 = (𝐹‘𝑦) → 𝑋 ∈ 𝑧))
11	10	albii 1839	. . . . . 6 ⊢ (∀𝑧((𝐹‘𝑦) = 𝑧 → 𝑋 ∈ 𝑧) ↔ ∀𝑧(𝑧 = (𝐹‘𝑦) → 𝑋 ∈ 𝑧))
12		fvex 6880	. . . . . . 7 ⊢ (𝐹‘𝑦) ∈ V
13		eleq2 2851	. . . . . . 7 ⊢ (𝑧 = (𝐹‘𝑦) → (𝑋 ∈ 𝑧 ↔ 𝑋 ∈ (𝐹‘𝑦)))
14	12, 13	ceqsalv 3493	. . . . . 6 ⊢ (∀𝑧(𝑧 = (𝐹‘𝑦) → 𝑋 ∈ 𝑧) ↔ 𝑋 ∈ (𝐹‘𝑦))
15	11, 14	bitri 277	. . . . 5 ⊢ (∀𝑧((𝐹‘𝑦) = 𝑧 → 𝑋 ∈ 𝑧) ↔ 𝑋 ∈ (𝐹‘𝑦))
16	15	ralbii 3108	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧((𝐹‘𝑦) = 𝑧 → 𝑋 ∈ 𝑧) ↔ ∀𝑦 ∈ 𝐵 𝑋 ∈ (𝐹‘𝑦))
17	8, 16	bitr3i 279	. . 3 ⊢ (∀𝑧∀𝑦 ∈ 𝐵 ((𝐹‘𝑦) = 𝑧 → 𝑋 ∈ 𝑧) ↔ ∀𝑦 ∈ 𝐵 𝑋 ∈ (𝐹‘𝑦))
18	7, 17	bitrdi 289	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∀𝑧(𝑧 ∈ (𝐹 “ 𝐵) → 𝑋 ∈ 𝑧) ↔ ∀𝑦 ∈ 𝐵 𝑋 ∈ (𝐹‘𝑦)))
19	2, 18	bitrid 285	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝑋 ∈ ∩ (𝐹 “ 𝐵) ↔ ∀𝑦 ∈ 𝐵 𝑋 ∈ (𝐹‘𝑦)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399  ∀wal 1558   = wceq 1560   ∈ wcel 2142  ∀wral 3076  ∃wrex 3086  Vcvv 3454   ⊆ wss 3904  ∩ cint 4905   “ cima 5650   Fn wfn 6516  ‘cfv 6521

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5246  ax-nul 5256  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-nf 1804  df-sb 2091  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4906  df-br 5101  df-opab 5163  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-res 5659  df-ima 5660  df-iota 6477  df-fun 6523  df-fn 6524  df-fv 6529

This theorem is referenced by: (None)