Theorem iinexg 5304

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 4988	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵})
2	1	adantl 485	. 2 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵})
3		elisset 2844	. . . . . . . . 9 ⊢ (𝐵 ∈ 𝐶 → ∃𝑦 𝑦 = 𝐵)
4	3	rgenw 3080	. . . . . . . 8 ⊢ ∀𝑥 ∈ 𝐴 (𝐵 ∈ 𝐶 → ∃𝑦 𝑦 = 𝐵)
5		r19.2z 4453	. . . . . . . 8 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 (𝐵 ∈ 𝐶 → ∃𝑦 𝑦 = 𝐵)) → ∃𝑥 ∈ 𝐴 (𝐵 ∈ 𝐶 → ∃𝑦 𝑦 = 𝐵))
6	4, 5	mpan2 701	. . . . . . 7 ⊢ (𝐴 ≠ ∅ → ∃𝑥 ∈ 𝐴 (𝐵 ∈ 𝐶 → ∃𝑦 𝑦 = 𝐵))
7		r19.35 3120	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐴 (𝐵 ∈ 𝐶 → ∃𝑦 𝑦 = 𝐵) ↔ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ∃𝑥 ∈ 𝐴 ∃𝑦 𝑦 = 𝐵))
8	6, 7	sylib 220	. . . . . 6 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ∃𝑥 ∈ 𝐴 ∃𝑦 𝑦 = 𝐵))
9	8	imp 410	. . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) → ∃𝑥 ∈ 𝐴 ∃𝑦 𝑦 = 𝐵)
10		rexcom4 3289	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 𝑦 = 𝐵 ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝑦 = 𝐵)
11	9, 10	sylib 220	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) → ∃𝑦∃𝑥 ∈ 𝐴 𝑦 = 𝐵)
12		abn0 4338	. . . 4 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ≠ ∅ ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝑦 = 𝐵)
13	11, 12	sylibr 236	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ≠ ∅)
14		intex 5300	. . 3 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ≠ ∅ ↔ ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V)
15	13, 14	sylib 220	. 2 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) → ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V)
16	2, 15	eqeltrd 2862	1 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ V)

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1560  ∃wex 1799   ∈ wcel 2142  {cab 2740   ≠ wne 2957  ∀wral 3076  ∃wrex 3086  Vcvv 3454  ∅c0 4285  ∩ cint 4905  ∩ ciin 4950

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

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-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-in 3911  df-ss 3921  df-nul 4286  df-int 4906  df-iin 4952

This theorem is referenced by:  fclsval  24068  taylfval  26422  iinexd  45711  smflimlem1  47345  smfliminflem  47404  iinfssclem2  49676  iinfssclem3  49677  iinfssc  49678