Theorem iinexgm 4172

Description: The existence of an indexed union. 𝑥 is normally a free-variable parameter in 𝐵, which should be read 𝐵(𝑥). (Contributed by Jim Kingdon, 28-Aug-2018.)

Assertion

Ref	Expression
iinexgm	⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ V)

Distinct variable group:   𝑥,𝐴

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

Proof of Theorem iinexgm

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		dfiin2g 3934	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵})
2	1	adantl 277	. 2 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) → ∩ 𝑥 ∈ 𝐴 𝐵 = ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵})
3		elisset 2766	. . . . . . . . . 10 ⊢ (𝐵 ∈ 𝐶 → ∃𝑦 𝑦 = 𝐵)
4	3	rgenw 2545	. . . . . . . . 9 ⊢ ∀𝑥 ∈ 𝐴 (𝐵 ∈ 𝐶 → ∃𝑦 𝑦 = 𝐵)
5		r19.2m 3524	. . . . . . . . 9 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐵 ∈ 𝐶 → ∃𝑦 𝑦 = 𝐵)) → ∃𝑥 ∈ 𝐴 (𝐵 ∈ 𝐶 → ∃𝑦 𝑦 = 𝐵))
6	4, 5	mpan2 425	. . . . . . . 8 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 (𝐵 ∈ 𝐶 → ∃𝑦 𝑦 = 𝐵))
7		r19.35-1 2640	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐴 (𝐵 ∈ 𝐶 → ∃𝑦 𝑦 = 𝐵) → (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ∃𝑥 ∈ 𝐴 ∃𝑦 𝑦 = 𝐵))
8	6, 7	syl 14	. . . . . . 7 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ∃𝑥 ∈ 𝐴 ∃𝑦 𝑦 = 𝐵))
9	8	imp 124	. . . . . 6 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) → ∃𝑥 ∈ 𝐴 ∃𝑦 𝑦 = 𝐵)
10		rexcom4 2775	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦 𝑦 = 𝐵 ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝑦 = 𝐵)
11	9, 10	sylib 122	. . . . 5 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) → ∃𝑦∃𝑥 ∈ 𝐴 𝑦 = 𝐵)
12		abid 2177	. . . . . 6 ⊢ (𝑦 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ↔ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵)
13	12	exbii 1616	. . . . 5 ⊢ (∃𝑦 𝑦 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ↔ ∃𝑦∃𝑥 ∈ 𝐴 𝑦 = 𝐵)
14	11, 13	sylibr 134	. . . 4 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) → ∃𝑦 𝑦 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵})
15		nfv 1539	. . . . 5 ⊢ Ⅎ𝑧 𝑦 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}
16		nfsab1 2179	. . . . 5 ⊢ Ⅎ𝑦 𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}
17		eleq1w 2250	. . . . 5 ⊢ (𝑦 = 𝑧 → (𝑦 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ↔ 𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}))
18	15, 16, 17	cbvex 1767	. . . 4 ⊢ (∃𝑦 𝑦 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ↔ ∃𝑧 𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵})
19	14, 18	sylib 122	. . 3 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) → ∃𝑧 𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵})
20		inteximm 4167	. . 3 ⊢ (∃𝑧 𝑧 ∈ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} → ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V)
21	19, 20	syl 14	. 2 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) → ∩ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ∈ V)
22	2, 21	eqeltrd 2266	1 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) → ∩ 𝑥 ∈ 𝐴 𝐵 ∈ V)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364  ∃wex 1503   ∈ wcel 2160  {cab 2175  ∀wral 2468  ∃wrex 2469  Vcvv 2752  ∩ cint 3859  ∩ ciin 3902

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-sep 4136

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-in 3150  df-ss 3157  df-int 3860  df-iin 3904

This theorem is referenced by: (None)