Theorem elintfv 31662

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 4481	. 2 ⊢ (𝑋 ∈ ∩ (𝐹 “ 𝐵) ↔ ∀𝑧(𝑧 ∈ (𝐹 “ 𝐵) → 𝑋 ∈ 𝑧))
3		ssel2 3598	. . . . . . . . . 10 ⊢ ((𝐵 ⊆ 𝐴 ∧ 𝑦 ∈ 𝐵) → 𝑦 ∈ 𝐴)
4		fnbrfvb 6236	. . . . . . . . . 10 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝐹‘𝑦) = 𝑧 ↔ 𝑦𝐹𝑧))
5	3, 4	sylan2 491	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ (𝐵 ⊆ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ((𝐹‘𝑦) = 𝑧 ↔ 𝑦𝐹𝑧))
6	5	anassrs 680	. . . . . . . 8 ⊢ (((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) ∧ 𝑦 ∈ 𝐵) → ((𝐹‘𝑦) = 𝑧 ↔ 𝑦𝐹𝑧))
7	6	rexbidva 3049	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑧 ↔ ∃𝑦 ∈ 𝐵 𝑦𝐹𝑧))
8		vex 3203	. . . . . . . 8 ⊢ 𝑧 ∈ V
9	8	elima 5471	. . . . . . 7 ⊢ (𝑧 ∈ (𝐹 “ 𝐵) ↔ ∃𝑦 ∈ 𝐵 𝑦𝐹𝑧)
10	7, 9	syl6rbbr 279	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝑧 ∈ (𝐹 “ 𝐵) ↔ ∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑧))
11	10	imbi1d 331	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → ((𝑧 ∈ (𝐹 “ 𝐵) → 𝑋 ∈ 𝑧) ↔ (∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑧 → 𝑋 ∈ 𝑧)))
12		r19.23v 3023	. . . . 5 ⊢ (∀𝑦 ∈ 𝐵 ((𝐹‘𝑦) = 𝑧 → 𝑋 ∈ 𝑧) ↔ (∃𝑦 ∈ 𝐵 (𝐹‘𝑦) = 𝑧 → 𝑋 ∈ 𝑧))
13	11, 12	syl6bbr 278	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → ((𝑧 ∈ (𝐹 “ 𝐵) → 𝑋 ∈ 𝑧) ↔ ∀𝑦 ∈ 𝐵 ((𝐹‘𝑦) = 𝑧 → 𝑋 ∈ 𝑧)))
14	13	albidv 1849	. . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∀𝑧(𝑧 ∈ (𝐹 “ 𝐵) → 𝑋 ∈ 𝑧) ↔ ∀𝑧∀𝑦 ∈ 𝐵 ((𝐹‘𝑦) = 𝑧 → 𝑋 ∈ 𝑧)))
15		ralcom4 3224	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧((𝐹‘𝑦) = 𝑧 → 𝑋 ∈ 𝑧) ↔ ∀𝑧∀𝑦 ∈ 𝐵 ((𝐹‘𝑦) = 𝑧 → 𝑋 ∈ 𝑧))
16		eqcom 2629	. . . . . . . 8 ⊢ ((𝐹‘𝑦) = 𝑧 ↔ 𝑧 = (𝐹‘𝑦))
17	16	imbi1i 339	. . . . . . 7 ⊢ (((𝐹‘𝑦) = 𝑧 → 𝑋 ∈ 𝑧) ↔ (𝑧 = (𝐹‘𝑦) → 𝑋 ∈ 𝑧))
18	17	albii 1747	. . . . . 6 ⊢ (∀𝑧((𝐹‘𝑦) = 𝑧 → 𝑋 ∈ 𝑧) ↔ ∀𝑧(𝑧 = (𝐹‘𝑦) → 𝑋 ∈ 𝑧))
19		fvex 6201	. . . . . . 7 ⊢ (𝐹‘𝑦) ∈ V
20		eleq2 2690	. . . . . . 7 ⊢ (𝑧 = (𝐹‘𝑦) → (𝑋 ∈ 𝑧 ↔ 𝑋 ∈ (𝐹‘𝑦)))
21	19, 20	ceqsalv 3233	. . . . . 6 ⊢ (∀𝑧(𝑧 = (𝐹‘𝑦) → 𝑋 ∈ 𝑧) ↔ 𝑋 ∈ (𝐹‘𝑦))
22	18, 21	bitri 264	. . . . 5 ⊢ (∀𝑧((𝐹‘𝑦) = 𝑧 → 𝑋 ∈ 𝑧) ↔ 𝑋 ∈ (𝐹‘𝑦))
23	22	ralbii 2980	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 ∀𝑧((𝐹‘𝑦) = 𝑧 → 𝑋 ∈ 𝑧) ↔ ∀𝑦 ∈ 𝐵 𝑋 ∈ (𝐹‘𝑦))
24	15, 23	bitr3i 266	. . 3 ⊢ (∀𝑧∀𝑦 ∈ 𝐵 ((𝐹‘𝑦) = 𝑧 → 𝑋 ∈ 𝑧) ↔ ∀𝑦 ∈ 𝐵 𝑋 ∈ (𝐹‘𝑦))
25	14, 24	syl6bb 276	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (∀𝑧(𝑧 ∈ (𝐹 “ 𝐵) → 𝑋 ∈ 𝑧) ↔ ∀𝑦 ∈ 𝐵 𝑋 ∈ (𝐹‘𝑦)))
26	2, 25	syl5bb 272	1 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐵 ⊆ 𝐴) → (𝑋 ∈ ∩ (𝐹 “ 𝐵) ↔ ∀𝑦 ∈ 𝐵 𝑋 ∈ (𝐹‘𝑦)))

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384  ∀wal 1481   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913  Vcvv 3200   ⊆ wss 3574  ∩ cint 4475   class class class wbr 4653   “ cima 5117   Fn wfn 5883  ‘cfv 5888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896

This theorem is referenced by: (None)