Theorem iinexg 5309

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 4989	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵})
2	1	adantl 487	. 2 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵})
3		elisset 2843	. . . . . . . . 9 ⊢ (𝐵 ∈ 𝐶 → ∃𝑦 𝑦 = 𝐵)
4	3	rgenw 3081	. . . . . . . 8 ⊢ ∀𝑥 ∈ 𝐴 (𝐵 ∈ 𝐶 → ∃𝑦 𝑦 = 𝐵)
5		r19.2z 4455	. . . . . . . 8 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 (𝐵 ∈ 𝐶 → ∃𝑦 𝑦 = 𝐵)) → ∃𝑥 ∈ 𝐴 (𝐵 ∈ 𝐶 → ∃𝑦 𝑦 = 𝐵))
6	4, 5	mpan2 704	. . . . . . 7 ⊢ (𝐴 ≠ ∅ → ∃𝑥 ∈ 𝐴 (𝐵 ∈ 𝐶 → ∃𝑦 𝑦 = 𝐵))
7		r19.35 3121	. . . . . . 7 ⊢ (∃𝑥 ∈ 𝐴 (𝐵 ∈ 𝐶 → ∃𝑦 𝑦 = 𝐵) ↔ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ∃𝑥 ∈ 𝐴 ∃𝑦 𝑦 = 𝐵))
8	6, 7	sylib 221	. . . . . 6 ⊢ (𝐴 ≠ ∅ → (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ∃𝑥 ∈ 𝐴 ∃𝑦 𝑦 = 𝐵))
9	8	imp 412	. . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) → ∃𝑥 ∈ 𝐴 ∃𝑦 𝑦 = 𝐵)
10		rexcom4 3290	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 𝑦 = 𝐵 ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝑦 = 𝐵)
11	9, 10	sylib 221	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) → ∃𝑦∃𝑥 ∈ 𝐴 𝑦 = 𝐵)
12		abn0 4334	. . . 4 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ≠ ∅ ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝑦 = 𝐵)
13	11, 12	sylibr 237	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ≠ ∅)
14		intex 5305	. . 3 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ≠ ∅ ↔ ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V)
15	13, 14	sylib 221	. 2 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) → ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V)
16	2, 15	eqeltrd 2861	1 ⊢ ((𝐴 ≠ ∅ ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ V)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  {cab 2739   ≠ wne 2956  ∀wral 3077  ∃wrex 3087  Vcvv 3451  ∅c0 4279  ∩ cint 4907  ∩ ciin 4952

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-in 3906  df-ss 3916  df-nul 4280  df-int 4908  df-iin 4954

This theorem is used by:  fclsval  24327  taylfval  26686  iinexd  46147  smflimlem1  47780  smfliminflem  47839  iinfssclem2  50162  iinfssclem3  50163  iinfssc  50164