Theorem iinexg 5274

Description: The existence of a class intersection. 𝑥 is normally a free-variable parameter in 𝐵, which should be read 𝐵(𝑥). (Contributed by FL, 19-Sep-2011.)

Assertion

Ref	Expression
iinexg	⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ V)

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)

Proof of Theorem iinexg

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfiin2g 4969	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵})
2	1	adantl 483	. 2 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵})
3		elisset 2818	. . . . . . . . 9 ⊢ (𝐵 ∈ 𝐶 → ∃𝑦 𝑦 = 𝐵)
4	3	rgenw 3066	. . . . . . . 8 ⊢ ∀𝑥 ∈ 𝐴 (𝐵 ∈ 𝐶 → ∃𝑦 𝑦 = 𝐵)
5		r19.2z 4431	. . . . . . . 8 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 (𝐵 ∈ 𝐶 → ∃𝑦 𝑦 = 𝐵)) → ∃𝑥 ∈ 𝐴 (𝐵 ∈ 𝐶 → ∃𝑦 𝑦 = 𝐵))
6	4, 5	mpan2 689	. . . . . . 7 ⊢ (𝐴 ≠ ∅ → ∃𝑥 ∈ 𝐴 (𝐵 ∈ 𝐶 → ∃𝑦 𝑦 = 𝐵))
7		r19.35 3108	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐴 (𝐵 ∈ 𝐶 → ∃𝑦 𝑦 = 𝐵) ↔ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ∃𝑥 ∈ 𝐴 ∃𝑦 𝑦 = 𝐵))
8	6, 7	sylib 217	. . . . . 6 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ∃𝑥 ∈ 𝐴 ∃𝑦 𝑦 = 𝐵))
9	8	imp 408	. . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) → ∃𝑥 ∈ 𝐴 ∃𝑦 𝑦 = 𝐵)
10		rexcom4 3268	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 𝑦 = 𝐵 ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝑦 = 𝐵)
11	9, 10	sylib 217	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) → ∃𝑦∃𝑥 ∈ 𝐴 𝑦 = 𝐵)
12		abn0 4320	. . . 4 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ≠ ∅ ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝑦 = 𝐵)
13	11, 12	sylibr 233	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ≠ ∅)
14		intex 5270	. . 3 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ≠ ∅ ↔ ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V)
15	13, 14	sylib 217	. 2 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) → ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V)
16	2, 15	eqeltrd 2837	1 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ V)

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1539  ∃wex 1779   ∈ wcel 2104  {cab 2713   ≠ wne 2941  ∀wral 3062  ∃wrex 3071  Vcvv 3437  ∅c0 4262  ∩ cint 4886  ∩ ciin 4932

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-sep 5232

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3287  df-v 3439  df-dif 3895  df-in 3899  df-ss 3909  df-nul 4263  df-int 4887  df-iin 4934

This theorem is referenced by:  fclsval  23204  taylfval  25563  iinexd  42720  smflimlem1  44359  smfliminflem  44417