Theorem bnj865 34954

Description: Technical lemma for bnj69 35041. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj865.1	⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj865.2	⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj865.3	⊢ 𝐷 = (ω ∖ {∅})
bnj865.5	⊢ (𝜒 ↔ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑛 ∈ 𝐷))
bnj865.6	⊢ (𝜃 ↔ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))

Assertion

Ref	Expression
bnj865	⊢ ∃𝑤∀𝑛(𝜒 → ∃𝑓 ∈ 𝑤 𝜃)

Distinct variable groups:   𝐴,𝑓,𝑖,𝑛,𝑦   𝑤,𝐴,𝑓,𝑛   𝐷,𝑓,𝑖,𝑛   𝑤,𝐷   𝑅,𝑓,𝑖,𝑛,𝑦   𝑤,𝑅   𝑓,𝑋,𝑛,𝑤   𝜑,𝑤   𝜓,𝑤

Allowed substitution hints:   𝜑(𝑦,𝑓,𝑖,𝑛)   𝜓(𝑦,𝑓,𝑖,𝑛)   𝜒(𝑦,𝑤,𝑓,𝑖,𝑛)   𝜃(𝑦,𝑤,𝑓,𝑖,𝑛)   𝐷(𝑦)   𝑋(𝑦,𝑖)

Proof of Theorem bnj865

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj865.1	. . . . . . 7 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
2		bnj865.2	. . . . . . 7 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
3		bnj865.3	. . . . . . 7 ⊢ 𝐷 = (ω ∖ {∅})
4	1, 2, 3	bnj852 34952	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∀𝑛 ∈ 𝐷 ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
5		omex 9657	. . . . . . . . 9 ⊢ ω ∈ V
6		difexg 5299	. . . . . . . . 9 ⊢ (ω ∈ V → (ω ∖ {∅}) ∈ V)
7	5, 6	ax-mp 5	. . . . . . . 8 ⊢ (ω ∖ {∅}) ∈ V
8	3, 7	eqeltri 2830	. . . . . . 7 ⊢ 𝐷 ∈ V
9		raleq 3302	. . . . . . . 8 ⊢ (𝑧 = 𝐷 → (∀𝑛 ∈ 𝑧 ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ ∀𝑛 ∈ 𝐷 ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
10		raleq 3302	. . . . . . . . 9 ⊢ (𝑧 = 𝐷 → (∀𝑛 ∈ 𝑧 ∃𝑓 ∈ 𝑤 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ ∀𝑛 ∈ 𝐷 ∃𝑓 ∈ 𝑤 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
11	10	exbidv 1921	. . . . . . . 8 ⊢ (𝑧 = 𝐷 → (∃𝑤∀𝑛 ∈ 𝑧 ∃𝑓 ∈ 𝑤 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ ∃𝑤∀𝑛 ∈ 𝐷 ∃𝑓 ∈ 𝑤 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
12	9, 11	imbi12d 344	. . . . . . 7 ⊢ (𝑧 = 𝐷 → ((∀𝑛 ∈ 𝑧 ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) → ∃𝑤∀𝑛 ∈ 𝑧 ∃𝑓 ∈ 𝑤 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) ↔ (∀𝑛 ∈ 𝐷 ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) → ∃𝑤∀𝑛 ∈ 𝐷 ∃𝑓 ∈ 𝑤 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))))
13		zfrep6 7953	. . . . . . 7 ⊢ (∀𝑛 ∈ 𝑧 ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) → ∃𝑤∀𝑛 ∈ 𝑧 ∃𝑓 ∈ 𝑤 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
14	8, 12, 13	vtocl 3537	. . . . . 6 ⊢ (∀𝑛 ∈ 𝐷 ∃!𝑓(𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) → ∃𝑤∀𝑛 ∈ 𝐷 ∃𝑓 ∈ 𝑤 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
15	4, 14	syl 17	. . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∃𝑤∀𝑛 ∈ 𝐷 ∃𝑓 ∈ 𝑤 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
16		19.37v 1991	. . . . 5 ⊢ (∃𝑤((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∀𝑛 ∈ 𝐷 ∃𝑓 ∈ 𝑤 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∃𝑤∀𝑛 ∈ 𝐷 ∃𝑓 ∈ 𝑤 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
17	15, 16	mpbir 231	. . . 4 ⊢ ∃𝑤((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∀𝑛 ∈ 𝐷 ∃𝑓 ∈ 𝑤 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
18		df-ral 3052	. . . . . . . 8 ⊢ (∀𝑛 ∈ 𝐷 ∃𝑓 ∈ 𝑤 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ ∀𝑛(𝑛 ∈ 𝐷 → ∃𝑓 ∈ 𝑤 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
19	18	imbi2i 336	. . . . . . 7 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∀𝑛 ∈ 𝐷 ∃𝑓 ∈ 𝑤 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∀𝑛(𝑛 ∈ 𝐷 → ∃𝑓 ∈ 𝑤 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))))
20		19.21v 1939	. . . . . . 7 ⊢ (∀𝑛((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑛 ∈ 𝐷 → ∃𝑓 ∈ 𝑤 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∀𝑛(𝑛 ∈ 𝐷 → ∃𝑓 ∈ 𝑤 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))))
21	19, 20	bitr4i 278	. . . . . 6 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∀𝑛 ∈ 𝐷 ∃𝑓 ∈ 𝑤 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) ↔ ∀𝑛((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑛 ∈ 𝐷 → ∃𝑓 ∈ 𝑤 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))))
22	21	exbii 1848	. . . . 5 ⊢ (∃𝑤((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∀𝑛 ∈ 𝐷 ∃𝑓 ∈ 𝑤 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) ↔ ∃𝑤∀𝑛((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑛 ∈ 𝐷 → ∃𝑓 ∈ 𝑤 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))))
23		impexp 450	. . . . . . . 8 ⊢ ((((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑛 ∈ 𝐷) → ∃𝑓 ∈ 𝑤 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑛 ∈ 𝐷 → ∃𝑓 ∈ 𝑤 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))))
24		df-3an 1088	. . . . . . . . . 10 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑛 ∈ 𝐷) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑛 ∈ 𝐷))
25	24	bicomi 224	. . . . . . . . 9 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑛 ∈ 𝐷) ↔ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑛 ∈ 𝐷))
26	25	imbi1i 349	. . . . . . . 8 ⊢ ((((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑛 ∈ 𝐷) → ∃𝑓 ∈ 𝑤 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑛 ∈ 𝐷) → ∃𝑓 ∈ 𝑤 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
27	23, 26	bitr3i 277	. . . . . . 7 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑛 ∈ 𝐷 → ∃𝑓 ∈ 𝑤 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑛 ∈ 𝐷) → ∃𝑓 ∈ 𝑤 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
28	27	albii 1819	. . . . . 6 ⊢ (∀𝑛((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑛 ∈ 𝐷 → ∃𝑓 ∈ 𝑤 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))) ↔ ∀𝑛((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑛 ∈ 𝐷) → ∃𝑓 ∈ 𝑤 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
29	28	exbii 1848	. . . . 5 ⊢ (∃𝑤∀𝑛((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝑛 ∈ 𝐷 → ∃𝑓 ∈ 𝑤 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))) ↔ ∃𝑤∀𝑛((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑛 ∈ 𝐷) → ∃𝑓 ∈ 𝑤 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
30	22, 29	bitri 275	. . . 4 ⊢ (∃𝑤((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → ∀𝑛 ∈ 𝐷 ∃𝑓 ∈ 𝑤 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) ↔ ∃𝑤∀𝑛((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑛 ∈ 𝐷) → ∃𝑓 ∈ 𝑤 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
31	17, 30	mpbi 230	. . 3 ⊢ ∃𝑤∀𝑛((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑛 ∈ 𝐷) → ∃𝑓 ∈ 𝑤 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
32		bnj865.5	. . . . . . 7 ⊢ (𝜒 ↔ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑛 ∈ 𝐷))
33	32	bicomi 224	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑛 ∈ 𝐷) ↔ 𝜒)
34	33	imbi1i 349	. . . . 5 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑛 ∈ 𝐷) → ∃𝑓 ∈ 𝑤 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) ↔ (𝜒 → ∃𝑓 ∈ 𝑤 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
35	34	albii 1819	. . . 4 ⊢ (∀𝑛((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑛 ∈ 𝐷) → ∃𝑓 ∈ 𝑤 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) ↔ ∀𝑛(𝜒 → ∃𝑓 ∈ 𝑤 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
36	35	exbii 1848	. . 3 ⊢ (∃𝑤∀𝑛((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑛 ∈ 𝐷) → ∃𝑓 ∈ 𝑤 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)) ↔ ∃𝑤∀𝑛(𝜒 → ∃𝑓 ∈ 𝑤 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
37	31, 36	mpbi 230	. 2 ⊢ ∃𝑤∀𝑛(𝜒 → ∃𝑓 ∈ 𝑤 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
38		bnj865.6	. . . . . 6 ⊢ (𝜃 ↔ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
39	38	rexbii 3083	. . . . 5 ⊢ (∃𝑓 ∈ 𝑤 𝜃 ↔ ∃𝑓 ∈ 𝑤 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
40	39	imbi2i 336	. . . 4 ⊢ ((𝜒 → ∃𝑓 ∈ 𝑤 𝜃) ↔ (𝜒 → ∃𝑓 ∈ 𝑤 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
41	40	albii 1819	. . 3 ⊢ (∀𝑛(𝜒 → ∃𝑓 ∈ 𝑤 𝜃) ↔ ∀𝑛(𝜒 → ∃𝑓 ∈ 𝑤 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
42	41	exbii 1848	. 2 ⊢ (∃𝑤∀𝑛(𝜒 → ∃𝑓 ∈ 𝑤 𝜃) ↔ ∃𝑤∀𝑛(𝜒 → ∃𝑓 ∈ 𝑤 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)))
43	37, 42	mpbir 231	1 ⊢ ∃𝑤∀𝑛(𝜒 → ∃𝑓 ∈ 𝑤 𝜃)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086  ∀wal 1538   = wceq 1540  ∃wex 1779   ∈ wcel 2108  ∃!weu 2567  ∀wral 3051  ∃wrex 3060  Vcvv 3459   ∖ cdif 3923  ∅c0 4308  {csn 4601  ∪ ciun 4967  suc csuc 6354   Fn wfn 6526  ‘cfv 6531  ωcom 7861   predc-bnj14 34719   FrSe w-bnj15 34723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729  ax-reg 9606  ax-inf2 9655

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-pss 3946  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-tr 5230  df-id 5548  df-eprel 5553  df-po 5561  df-so 5562  df-fr 5606  df-we 5608  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-om 7862  df-1o 8480  df-bnj17 34718  df-bnj14 34720  df-bnj13 34722  df-bnj15 34724

This theorem is referenced by:  bnj849  34956