Theorem iinexg 5342

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 5036	. . 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 2816	. . . . . . . . 9 ⊢ (𝐵 ∈ 𝐶 → ∃𝑦 𝑦 = 𝐵)
4	3	rgenw 3066	. . . . . . . 8 ⊢ ∀𝑥 ∈ 𝐴 (𝐵 ∈ 𝐶 → ∃𝑦 𝑦 = 𝐵)
5		r19.2z 4495	. . . . . . . 8 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 (𝐵 ∈ 𝐶 → ∃𝑦 𝑦 = 𝐵)) → ∃𝑥 ∈ 𝐴 (𝐵 ∈ 𝐶 → ∃𝑦 𝑦 = 𝐵))
6	4, 5	mpan2 690	. . . . . . 7 ⊢ (𝐴 ≠ ∅ → ∃𝑥 ∈ 𝐴 (𝐵 ∈ 𝐶 → ∃𝑦 𝑦 = 𝐵))
7		r19.35 3109	. . . . . . 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 3286	. . . . 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 4381	. . . 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 5338	. . 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 2834	1 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ V)

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1542  ∃wex 1782   ∈ wcel 2107  {cab 2710   ≠ wne 2941  ∀wral 3062  ∃wrex 3071  Vcvv 3475  ∅c0 4323  ∩ cint 4951  ∩ ciin 4999

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-in 3956  df-ss 3966  df-nul 4324  df-int 4952  df-iin 5001

This theorem is referenced by:  fclsval  23512  taylfval  25871  iinexd  43822  smflimlem1  45487  smfliminflem  45546