Theorem elmapintrab 40860

Description: Two ways to say a set is an element of the intersection of a class of images. (Contributed by RP, 16-Aug-2020.)

Hypotheses

Ref	Expression
elmapintrab.ex	⊢ 𝐶 ∈ V
elmapintrab.sub	⊢ 𝐶 ⊆ 𝐵

Assertion

Ref	Expression
elmapintrab	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ∩ {𝑤 ∈ 𝒫 𝐵 ∣ ∃𝑥(𝑤 = 𝐶 ∧ 𝜑)} ↔ ((∃𝑥𝜑 → 𝐴 ∈ 𝐵) ∧ ∀𝑥(𝜑 → 𝐴 ∈ 𝐶))))

Distinct variable groups:   𝜑,𝑤   𝑥,𝑤,𝐴   𝑤,𝐵,𝑥   𝑤,𝐶

Allowed substitution hints:   𝜑(𝑥)   𝐶(𝑥)   𝑉(𝑥,𝑤)

Proof of Theorem elmapintrab

Step	Hyp	Ref	Expression
1		elintrabg 4872	. . 3 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ∩ {𝑤 ∈ 𝒫 𝐵 ∣ ∃𝑥(𝑤 = 𝐶 ∧ 𝜑)} ↔ ∀𝑤 ∈ 𝒫 𝐵(∃𝑥(𝑤 = 𝐶 ∧ 𝜑) → 𝐴 ∈ 𝑤)))
2		df-ral 3066	. . 3 ⊢ (∀𝑤 ∈ 𝒫 𝐵(∃𝑥(𝑤 = 𝐶 ∧ 𝜑) → 𝐴 ∈ 𝑤) ↔ ∀𝑤(𝑤 ∈ 𝒫 𝐵 → (∃𝑥(𝑤 = 𝐶 ∧ 𝜑) → 𝐴 ∈ 𝑤)))
3	1, 2	bitrdi 290	. 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ∩ {𝑤 ∈ 𝒫 𝐵 ∣ ∃𝑥(𝑤 = 𝐶 ∧ 𝜑)} ↔ ∀𝑤(𝑤 ∈ 𝒫 𝐵 → (∃𝑥(𝑤 = 𝐶 ∧ 𝜑) → 𝐴 ∈ 𝑤))))
4		velpw 4518	. . . . . 6 ⊢ (𝑤 ∈ 𝒫 𝐵 ↔ 𝑤 ⊆ 𝐵)
5		19.23v 1950	. . . . . . 7 ⊢ (∀𝑥((𝑤 = 𝐶 ∧ 𝜑) → 𝐴 ∈ 𝑤) ↔ (∃𝑥(𝑤 = 𝐶 ∧ 𝜑) → 𝐴 ∈ 𝑤))
6	5	bicomi 227	. . . . . 6 ⊢ ((∃𝑥(𝑤 = 𝐶 ∧ 𝜑) → 𝐴 ∈ 𝑤) ↔ ∀𝑥((𝑤 = 𝐶 ∧ 𝜑) → 𝐴 ∈ 𝑤))
7	4, 6	imbi12i 354	. . . . 5 ⊢ ((𝑤 ∈ 𝒫 𝐵 → (∃𝑥(𝑤 = 𝐶 ∧ 𝜑) → 𝐴 ∈ 𝑤)) ↔ (𝑤 ⊆ 𝐵 → ∀𝑥((𝑤 = 𝐶 ∧ 𝜑) → 𝐴 ∈ 𝑤)))
8		19.21v 1947	. . . . 5 ⊢ (∀𝑥(𝑤 ⊆ 𝐵 → ((𝑤 = 𝐶 ∧ 𝜑) → 𝐴 ∈ 𝑤)) ↔ (𝑤 ⊆ 𝐵 → ∀𝑥((𝑤 = 𝐶 ∧ 𝜑) → 𝐴 ∈ 𝑤)))
9		bi2.04 392	. . . . . . 7 ⊢ ((𝑤 ⊆ 𝐵 → ((𝑤 = 𝐶 ∧ 𝜑) → 𝐴 ∈ 𝑤)) ↔ ((𝑤 = 𝐶 ∧ 𝜑) → (𝑤 ⊆ 𝐵 → 𝐴 ∈ 𝑤)))
10		impexp 454	. . . . . . 7 ⊢ (((𝑤 = 𝐶 ∧ 𝜑) → (𝑤 ⊆ 𝐵 → 𝐴 ∈ 𝑤)) ↔ (𝑤 = 𝐶 → (𝜑 → (𝑤 ⊆ 𝐵 → 𝐴 ∈ 𝑤))))
11	9, 10	bitri 278	. . . . . 6 ⊢ ((𝑤 ⊆ 𝐵 → ((𝑤 = 𝐶 ∧ 𝜑) → 𝐴 ∈ 𝑤)) ↔ (𝑤 = 𝐶 → (𝜑 → (𝑤 ⊆ 𝐵 → 𝐴 ∈ 𝑤))))
12	11	albii 1827	. . . . 5 ⊢ (∀𝑥(𝑤 ⊆ 𝐵 → ((𝑤 = 𝐶 ∧ 𝜑) → 𝐴 ∈ 𝑤)) ↔ ∀𝑥(𝑤 = 𝐶 → (𝜑 → (𝑤 ⊆ 𝐵 → 𝐴 ∈ 𝑤))))
13	7, 8, 12	3bitr2i 302	. . . 4 ⊢ ((𝑤 ∈ 𝒫 𝐵 → (∃𝑥(𝑤 = 𝐶 ∧ 𝜑) → 𝐴 ∈ 𝑤)) ↔ ∀𝑥(𝑤 = 𝐶 → (𝜑 → (𝑤 ⊆ 𝐵 → 𝐴 ∈ 𝑤))))
14	13	albii 1827	. . 3 ⊢ (∀𝑤(𝑤 ∈ 𝒫 𝐵 → (∃𝑥(𝑤 = 𝐶 ∧ 𝜑) → 𝐴 ∈ 𝑤)) ↔ ∀𝑤∀𝑥(𝑤 = 𝐶 → (𝜑 → (𝑤 ⊆ 𝐵 → 𝐴 ∈ 𝑤))))
15		alcom 2160	. . 3 ⊢ (∀𝑤∀𝑥(𝑤 = 𝐶 → (𝜑 → (𝑤 ⊆ 𝐵 → 𝐴 ∈ 𝑤))) ↔ ∀𝑥∀𝑤(𝑤 = 𝐶 → (𝜑 → (𝑤 ⊆ 𝐵 → 𝐴 ∈ 𝑤))))
16		elmapintrab.ex	. . . . . . 7 ⊢ 𝐶 ∈ V
17		sseq1 3926	. . . . . . . . 9 ⊢ (𝑤 = 𝐶 → (𝑤 ⊆ 𝐵 ↔ 𝐶 ⊆ 𝐵))
18		eleq2 2826	. . . . . . . . . 10 ⊢ (𝑤 = 𝐶 → (𝐴 ∈ 𝑤 ↔ 𝐴 ∈ 𝐶))
19		elmapintrab.sub	. . . . . . . . . . . 12 ⊢ 𝐶 ⊆ 𝐵
20	19	sseli 3896	. . . . . . . . . . 11 ⊢ (𝐴 ∈ 𝐶 → 𝐴 ∈ 𝐵)
21	20	pm4.71ri 564	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝐶 ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶))
22	18, 21	bitrdi 290	. . . . . . . . 9 ⊢ (𝑤 = 𝐶 → (𝐴 ∈ 𝑤 ↔ (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)))
23	17, 22	imbi12d 348	. . . . . . . 8 ⊢ (𝑤 = 𝐶 → ((𝑤 ⊆ 𝐵 → 𝐴 ∈ 𝑤) ↔ (𝐶 ⊆ 𝐵 → (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶))))
24	23	imbi2d 344	. . . . . . 7 ⊢ (𝑤 = 𝐶 → ((𝜑 → (𝑤 ⊆ 𝐵 → 𝐴 ∈ 𝑤)) ↔ (𝜑 → (𝐶 ⊆ 𝐵 → (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)))))
25	16, 24	ceqsalv 3443	. . . . . 6 ⊢ (∀𝑤(𝑤 = 𝐶 → (𝜑 → (𝑤 ⊆ 𝐵 → 𝐴 ∈ 𝑤))) ↔ (𝜑 → (𝐶 ⊆ 𝐵 → (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶))))
26		bi2.04 392	. . . . . 6 ⊢ ((𝜑 → (𝐶 ⊆ 𝐵 → (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶))) ↔ (𝐶 ⊆ 𝐵 → (𝜑 → (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶))))
27		pm5.5 365	. . . . . . . 8 ⊢ (𝐶 ⊆ 𝐵 → ((𝐶 ⊆ 𝐵 → (𝜑 → (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶))) ↔ (𝜑 → (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶))))
28	19, 27	ax-mp 5	. . . . . . 7 ⊢ ((𝐶 ⊆ 𝐵 → (𝜑 → (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶))) ↔ (𝜑 → (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)))
29		jcab 521	. . . . . . 7 ⊢ ((𝜑 → (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶)) ↔ ((𝜑 → 𝐴 ∈ 𝐵) ∧ (𝜑 → 𝐴 ∈ 𝐶)))
30	28, 29	bitri 278	. . . . . 6 ⊢ ((𝐶 ⊆ 𝐵 → (𝜑 → (𝐴 ∈ 𝐵 ∧ 𝐴 ∈ 𝐶))) ↔ ((𝜑 → 𝐴 ∈ 𝐵) ∧ (𝜑 → 𝐴 ∈ 𝐶)))
31	25, 26, 30	3bitri 300	. . . . 5 ⊢ (∀𝑤(𝑤 = 𝐶 → (𝜑 → (𝑤 ⊆ 𝐵 → 𝐴 ∈ 𝑤))) ↔ ((𝜑 → 𝐴 ∈ 𝐵) ∧ (𝜑 → 𝐴 ∈ 𝐶)))
32	31	albii 1827	. . . 4 ⊢ (∀𝑥∀𝑤(𝑤 = 𝐶 → (𝜑 → (𝑤 ⊆ 𝐵 → 𝐴 ∈ 𝑤))) ↔ ∀𝑥((𝜑 → 𝐴 ∈ 𝐵) ∧ (𝜑 → 𝐴 ∈ 𝐶)))
33		19.26 1878	. . . 4 ⊢ (∀𝑥((𝜑 → 𝐴 ∈ 𝐵) ∧ (𝜑 → 𝐴 ∈ 𝐶)) ↔ (∀𝑥(𝜑 → 𝐴 ∈ 𝐵) ∧ ∀𝑥(𝜑 → 𝐴 ∈ 𝐶)))
34		19.23v 1950	. . . . 5 ⊢ (∀𝑥(𝜑 → 𝐴 ∈ 𝐵) ↔ (∃𝑥𝜑 → 𝐴 ∈ 𝐵))
35	34	anbi1i 627	. . . 4 ⊢ ((∀𝑥(𝜑 → 𝐴 ∈ 𝐵) ∧ ∀𝑥(𝜑 → 𝐴 ∈ 𝐶)) ↔ ((∃𝑥𝜑 → 𝐴 ∈ 𝐵) ∧ ∀𝑥(𝜑 → 𝐴 ∈ 𝐶)))
36	32, 33, 35	3bitri 300	. . 3 ⊢ (∀𝑥∀𝑤(𝑤 = 𝐶 → (𝜑 → (𝑤 ⊆ 𝐵 → 𝐴 ∈ 𝑤))) ↔ ((∃𝑥𝜑 → 𝐴 ∈ 𝐵) ∧ ∀𝑥(𝜑 → 𝐴 ∈ 𝐶)))
37	14, 15, 36	3bitri 300	. 2 ⊢ (∀𝑤(𝑤 ∈ 𝒫 𝐵 → (∃𝑥(𝑤 = 𝐶 ∧ 𝜑) → 𝐴 ∈ 𝑤)) ↔ ((∃𝑥𝜑 → 𝐴 ∈ 𝐵) ∧ ∀𝑥(𝜑 → 𝐴 ∈ 𝐶)))
38	3, 37	bitrdi 290	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ∩ {𝑤 ∈ 𝒫 𝐵 ∣ ∃𝑥(𝑤 = 𝐶 ∧ 𝜑)} ↔ ((∃𝑥𝜑 → 𝐴 ∈ 𝐵) ∧ ∀𝑥(𝜑 → 𝐴 ∈ 𝐶))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1541   = wceq 1543  ∃wex 1787   ∈ wcel 2110  ∀wral 3061  {crab 3065  Vcvv 3408   ⊆ wss 3866  𝒫 cpw 4513  ∩ cint 4859

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3066  df-rab 3070  df-v 3410  df-in 3873  df-ss 3883  df-pw 4515  df-int 4860

This theorem is referenced by:  elinintrab  40861  cnvcnvintabd  40884  cnvintabd  40887