Theorem intab 4934

Description: The intersection of a special case of a class abstraction. 𝑦 may be free in 𝜑 and 𝐴, which can be thought of a 𝜑(𝑦) and 𝐴(𝑦). Typically, abrexex2 7915 or abexssex 7916 can be used to satisfy the second hypothesis. (Contributed by NM, 28-Jul-2006.) (Proof shortened by Mario Carneiro, 14-Nov-2016.)

Hypotheses

Ref	Expression
intab.1	⊢ 𝐴 ∈ V
intab.2	⊢ {𝑥 ∣ ∃𝑦(𝜑 ∧ 𝑥 = 𝐴)} ∈ V

Assertion

Ref	Expression
intab	⊢ ∩ {𝑥 ∣ ∀𝑦(𝜑 → 𝐴 ∈ 𝑥)} = {𝑥 ∣ ∃𝑦(𝜑 ∧ 𝑥 = 𝐴)}

Distinct variable groups:   𝑥,𝐴   𝜑,𝑥   𝑥,𝑦

Allowed substitution hints:   𝜑(𝑦)   𝐴(𝑦)

Proof of Theorem intab

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2741	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → (𝑧 = 𝐴 ↔ 𝑥 = 𝐴))
2	1	anbi2d 631	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → ((𝜑 ∧ 𝑧 = 𝐴) ↔ (𝜑 ∧ 𝑥 = 𝐴)))
3	2	exbidv 1923	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → (∃𝑦(𝜑 ∧ 𝑧 = 𝐴) ↔ ∃𝑦(𝜑 ∧ 𝑥 = 𝐴)))
4	3	cbvabv 2807	. . . . . . 7 ⊢ {𝑧 ∣ ∃𝑦(𝜑 ∧ 𝑧 = 𝐴)} = {𝑥 ∣ ∃𝑦(𝜑 ∧ 𝑥 = 𝐴)}
5		intab.2	. . . . . . 7 ⊢ {𝑥 ∣ ∃𝑦(𝜑 ∧ 𝑥 = 𝐴)} ∈ V
6	4, 5	eqeltri 2833	. . . . . 6 ⊢ {𝑧 ∣ ∃𝑦(𝜑 ∧ 𝑧 = 𝐴)} ∈ V
7		nfe1 2156	. . . . . . . . 9 ⊢ Ⅎ𝑦∃𝑦(𝜑 ∧ 𝑧 = 𝐴)
8	7	nfab 2905	. . . . . . . 8 ⊢ Ⅎ𝑦{𝑧 ∣ ∃𝑦(𝜑 ∧ 𝑧 = 𝐴)}
9	8	nfeq2 2917	. . . . . . 7 ⊢ Ⅎ𝑦 𝑥 = {𝑧 ∣ ∃𝑦(𝜑 ∧ 𝑧 = 𝐴)}
10		eleq2 2826	. . . . . . . 8 ⊢ (𝑥 = {𝑧 ∣ ∃𝑦(𝜑 ∧ 𝑧 = 𝐴)} → (𝐴 ∈ 𝑥 ↔ 𝐴 ∈ {𝑧 ∣ ∃𝑦(𝜑 ∧ 𝑧 = 𝐴)}))
11	10	imbi2d 340	. . . . . . 7 ⊢ (𝑥 = {𝑧 ∣ ∃𝑦(𝜑 ∧ 𝑧 = 𝐴)} → ((𝜑 → 𝐴 ∈ 𝑥) ↔ (𝜑 → 𝐴 ∈ {𝑧 ∣ ∃𝑦(𝜑 ∧ 𝑧 = 𝐴)})))
12	9, 11	albid 2230	. . . . . 6 ⊢ (𝑥 = {𝑧 ∣ ∃𝑦(𝜑 ∧ 𝑧 = 𝐴)} → (∀𝑦(𝜑 → 𝐴 ∈ 𝑥) ↔ ∀𝑦(𝜑 → 𝐴 ∈ {𝑧 ∣ ∃𝑦(𝜑 ∧ 𝑧 = 𝐴)})))
13	6, 12	elab 3635	. . . . 5 ⊢ ({𝑧 ∣ ∃𝑦(𝜑 ∧ 𝑧 = 𝐴)} ∈ {𝑥 ∣ ∀𝑦(𝜑 → 𝐴 ∈ 𝑥)} ↔ ∀𝑦(𝜑 → 𝐴 ∈ {𝑧 ∣ ∃𝑦(𝜑 ∧ 𝑧 = 𝐴)}))
14		19.8a 2189	. . . . . . . . 9 ⊢ ((𝜑 ∧ 𝑧 = 𝐴) → ∃𝑦(𝜑 ∧ 𝑧 = 𝐴))
15	14	ex 412	. . . . . . . 8 ⊢ (𝜑 → (𝑧 = 𝐴 → ∃𝑦(𝜑 ∧ 𝑧 = 𝐴)))
16	15	alrimiv 1929	. . . . . . 7 ⊢ (𝜑 → ∀𝑧(𝑧 = 𝐴 → ∃𝑦(𝜑 ∧ 𝑧 = 𝐴)))
17		intab.1	. . . . . . . 8 ⊢ 𝐴 ∈ V
18	17	sbc6 3772	. . . . . . 7 ⊢ ([𝐴 / 𝑧]∃𝑦(𝜑 ∧ 𝑧 = 𝐴) ↔ ∀𝑧(𝑧 = 𝐴 → ∃𝑦(𝜑 ∧ 𝑧 = 𝐴)))
19	16, 18	sylibr 234	. . . . . 6 ⊢ (𝜑 → [𝐴 / 𝑧]∃𝑦(𝜑 ∧ 𝑧 = 𝐴))
20		df-sbc 3742	. . . . . 6 ⊢ ([𝐴 / 𝑧]∃𝑦(𝜑 ∧ 𝑧 = 𝐴) ↔ 𝐴 ∈ {𝑧 ∣ ∃𝑦(𝜑 ∧ 𝑧 = 𝐴)})
21	19, 20	sylib 218	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ {𝑧 ∣ ∃𝑦(𝜑 ∧ 𝑧 = 𝐴)})
22	13, 21	mpgbir 1801	. . . 4 ⊢ {𝑧 ∣ ∃𝑦(𝜑 ∧ 𝑧 = 𝐴)} ∈ {𝑥 ∣ ∀𝑦(𝜑 → 𝐴 ∈ 𝑥)}
23		intss1 4919	. . . 4 ⊢ ({𝑧 ∣ ∃𝑦(𝜑 ∧ 𝑧 = 𝐴)} ∈ {𝑥 ∣ ∀𝑦(𝜑 → 𝐴 ∈ 𝑥)} → ∩ {𝑥 ∣ ∀𝑦(𝜑 → 𝐴 ∈ 𝑥)} ⊆ {𝑧 ∣ ∃𝑦(𝜑 ∧ 𝑧 = 𝐴)})
24	22, 23	ax-mp 5	. . 3 ⊢ ∩ {𝑥 ∣ ∀𝑦(𝜑 → 𝐴 ∈ 𝑥)} ⊆ {𝑧 ∣ ∃𝑦(𝜑 ∧ 𝑧 = 𝐴)}
25		19.29r 1876	. . . . . . . 8 ⊢ ((∃𝑦(𝜑 ∧ 𝑧 = 𝐴) ∧ ∀𝑦(𝜑 → 𝐴 ∈ 𝑥)) → ∃𝑦((𝜑 ∧ 𝑧 = 𝐴) ∧ (𝜑 → 𝐴 ∈ 𝑥)))
26		simplr 769	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 = 𝐴) ∧ (𝜑 → 𝐴 ∈ 𝑥)) → 𝑧 = 𝐴)
27		pm3.35 803	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝜑 → 𝐴 ∈ 𝑥)) → 𝐴 ∈ 𝑥)
28	27	adantlr 716	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑧 = 𝐴) ∧ (𝜑 → 𝐴 ∈ 𝑥)) → 𝐴 ∈ 𝑥)
29	26, 28	eqeltrd 2837	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑧 = 𝐴) ∧ (𝜑 → 𝐴 ∈ 𝑥)) → 𝑧 ∈ 𝑥)
30	29	exlimiv 1932	. . . . . . . 8 ⊢ (∃𝑦((𝜑 ∧ 𝑧 = 𝐴) ∧ (𝜑 → 𝐴 ∈ 𝑥)) → 𝑧 ∈ 𝑥)
31	25, 30	syl 17	. . . . . . 7 ⊢ ((∃𝑦(𝜑 ∧ 𝑧 = 𝐴) ∧ ∀𝑦(𝜑 → 𝐴 ∈ 𝑥)) → 𝑧 ∈ 𝑥)
32	31	ex 412	. . . . . 6 ⊢ (∃𝑦(𝜑 ∧ 𝑧 = 𝐴) → (∀𝑦(𝜑 → 𝐴 ∈ 𝑥) → 𝑧 ∈ 𝑥))
33	32	alrimiv 1929	. . . . 5 ⊢ (∃𝑦(𝜑 ∧ 𝑧 = 𝐴) → ∀𝑥(∀𝑦(𝜑 → 𝐴 ∈ 𝑥) → 𝑧 ∈ 𝑥))
34		vex 3445	. . . . . 6 ⊢ 𝑧 ∈ V
35	34	elintab 4915	. . . . 5 ⊢ (𝑧 ∈ ∩ {𝑥 ∣ ∀𝑦(𝜑 → 𝐴 ∈ 𝑥)} ↔ ∀𝑥(∀𝑦(𝜑 → 𝐴 ∈ 𝑥) → 𝑧 ∈ 𝑥))
36	33, 35	sylibr 234	. . . 4 ⊢ (∃𝑦(𝜑 ∧ 𝑧 = 𝐴) → 𝑧 ∈ ∩ {𝑥 ∣ ∀𝑦(𝜑 → 𝐴 ∈ 𝑥)})
37	36	abssi 4021	. . 3 ⊢ {𝑧 ∣ ∃𝑦(𝜑 ∧ 𝑧 = 𝐴)} ⊆ ∩ {𝑥 ∣ ∀𝑦(𝜑 → 𝐴 ∈ 𝑥)}
38	24, 37	eqssi 3951	. 2 ⊢ ∩ {𝑥 ∣ ∀𝑦(𝜑 → 𝐴 ∈ 𝑥)} = {𝑧 ∣ ∃𝑦(𝜑 ∧ 𝑧 = 𝐴)}
39	38, 4	eqtri 2760	1 ⊢ ∩ {𝑥 ∣ ∀𝑦(𝜑 → 𝐴 ∈ 𝑥)} = {𝑥 ∣ ∃𝑦(𝜑 ∧ 𝑥 = 𝐴)}

Syntax hints:   → wi 4   ∧ wa 395  ∀wal 1540   = wceq 1542  ∃wex 1781   ∈ wcel 2114  {cab 2715  Vcvv 3441  [wsbc 3741   ⊆ wss 3902  ∩ cint 4903

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-nf 1786  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-v 3443  df-sbc 3742  df-ss 3919  df-int 4904

This theorem is referenced by: (None)