Theorem bnd2 4105

Description: A variant of the Boundedness Axiom bnd 4104 that picks a subset 𝑧 out of a possibly proper class 𝐵 in which a property is true. (Contributed by NM, 4-Feb-2004.)

Hypothesis

Ref	Expression
bnd2.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
bnd2	⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑧(𝑧 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑧 𝜑))

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

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

Proof of Theorem bnd2

Dummy variables 𝑤 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rex 2423	. . . 4 ⊢ (∃𝑦 ∈ 𝐵 𝜑 ↔ ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝜑))
2	1	ralbii 2444	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝜑))
3		bnd2.1	. . . 4 ⊢ 𝐴 ∈ V
4		raleq 2629	. . . . 5 ⊢ (𝑣 = 𝐴 → (∀𝑥 ∈ 𝑣 ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝜑) ↔ ∀𝑥 ∈ 𝐴 ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝜑)))
5		raleq 2629	. . . . . 6 ⊢ (𝑣 = 𝐴 → (∀𝑥 ∈ 𝑣 ∃𝑦 ∈ 𝑤 (𝑦 ∈ 𝐵 ∧ 𝜑) ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑤 (𝑦 ∈ 𝐵 ∧ 𝜑)))
6	5	exbidv 1798	. . . . 5 ⊢ (𝑣 = 𝐴 → (∃𝑤∀𝑥 ∈ 𝑣 ∃𝑦 ∈ 𝑤 (𝑦 ∈ 𝐵 ∧ 𝜑) ↔ ∃𝑤∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑤 (𝑦 ∈ 𝐵 ∧ 𝜑)))
7	4, 6	imbi12d 233	. . . 4 ⊢ (𝑣 = 𝐴 → ((∀𝑥 ∈ 𝑣 ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝜑) → ∃𝑤∀𝑥 ∈ 𝑣 ∃𝑦 ∈ 𝑤 (𝑦 ∈ 𝐵 ∧ 𝜑)) ↔ (∀𝑥 ∈ 𝐴 ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝜑) → ∃𝑤∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑤 (𝑦 ∈ 𝐵 ∧ 𝜑))))
8		bnd 4104	. . . 4 ⊢ (∀𝑥 ∈ 𝑣 ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝜑) → ∃𝑤∀𝑥 ∈ 𝑣 ∃𝑦 ∈ 𝑤 (𝑦 ∈ 𝐵 ∧ 𝜑))
9	3, 7, 8	vtocl 2743	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦(𝑦 ∈ 𝐵 ∧ 𝜑) → ∃𝑤∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑤 (𝑦 ∈ 𝐵 ∧ 𝜑))
10	2, 9	sylbi 120	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑤∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑤 (𝑦 ∈ 𝐵 ∧ 𝜑))
11		vex 2692	. . . . 5 ⊢ 𝑤 ∈ V
12	11	inex1 4070	. . . 4 ⊢ (𝑤 ∩ 𝐵) ∈ V
13		inss2 3302	. . . . . . 7 ⊢ (𝑤 ∩ 𝐵) ⊆ 𝐵
14		sseq1 3125	. . . . . . 7 ⊢ (𝑧 = (𝑤 ∩ 𝐵) → (𝑧 ⊆ 𝐵 ↔ (𝑤 ∩ 𝐵) ⊆ 𝐵))
15	13, 14	mpbiri 167	. . . . . 6 ⊢ (𝑧 = (𝑤 ∩ 𝐵) → 𝑧 ⊆ 𝐵)
16	15	biantrurd 303	. . . . 5 ⊢ (𝑧 = (𝑤 ∩ 𝐵) → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑧 𝜑 ↔ (𝑧 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑧 𝜑)))
17		rexeq 2630	. . . . . . 7 ⊢ (𝑧 = (𝑤 ∩ 𝐵) → (∃𝑦 ∈ 𝑧 𝜑 ↔ ∃𝑦 ∈ (𝑤 ∩ 𝐵)𝜑))
18		elin 3264	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝑤 ∩ 𝐵) ↔ (𝑦 ∈ 𝑤 ∧ 𝑦 ∈ 𝐵))
19	18	anbi1i 454	. . . . . . . . 9 ⊢ ((𝑦 ∈ (𝑤 ∩ 𝐵) ∧ 𝜑) ↔ ((𝑦 ∈ 𝑤 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑))
20		anass 399	. . . . . . . . 9 ⊢ (((𝑦 ∈ 𝑤 ∧ 𝑦 ∈ 𝐵) ∧ 𝜑) ↔ (𝑦 ∈ 𝑤 ∧ (𝑦 ∈ 𝐵 ∧ 𝜑)))
21	19, 20	bitri 183	. . . . . . . 8 ⊢ ((𝑦 ∈ (𝑤 ∩ 𝐵) ∧ 𝜑) ↔ (𝑦 ∈ 𝑤 ∧ (𝑦 ∈ 𝐵 ∧ 𝜑)))
22	21	rexbii2 2449	. . . . . . 7 ⊢ (∃𝑦 ∈ (𝑤 ∩ 𝐵)𝜑 ↔ ∃𝑦 ∈ 𝑤 (𝑦 ∈ 𝐵 ∧ 𝜑))
23	17, 22	syl6bb 195	. . . . . 6 ⊢ (𝑧 = (𝑤 ∩ 𝐵) → (∃𝑦 ∈ 𝑧 𝜑 ↔ ∃𝑦 ∈ 𝑤 (𝑦 ∈ 𝐵 ∧ 𝜑)))
24	23	ralbidv 2438	. . . . 5 ⊢ (𝑧 = (𝑤 ∩ 𝐵) → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑧 𝜑 ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑤 (𝑦 ∈ 𝐵 ∧ 𝜑)))
25	16, 24	bitr3d 189	. . . 4 ⊢ (𝑧 = (𝑤 ∩ 𝐵) → ((𝑧 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑧 𝜑) ↔ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑤 (𝑦 ∈ 𝐵 ∧ 𝜑)))
26	12, 25	spcev 2784	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑤 (𝑦 ∈ 𝐵 ∧ 𝜑) → ∃𝑧(𝑧 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑧 𝜑))
27	26	exlimiv 1578	. 2 ⊢ (∃𝑤∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑤 (𝑦 ∈ 𝐵 ∧ 𝜑) → ∃𝑧(𝑧 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑧 𝜑))
28	10, 27	syl 14	1 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝜑 → ∃𝑧(𝑧 ⊆ 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝑧 𝜑))

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1332  ∃wex 1469   ∈ wcel 1481  ∀wral 2417  ∃wrex 2418  Vcvv 2689   ∩ cin 3075   ⊆ wss 3076

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4051  ax-sep 4054

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-in 3082  df-ss 3089

This theorem is referenced by: (None)