Theorem bnj1033 35666

Description: Technical lemma for bnj69 35707. 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
bnj1033.1	⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj1033.2	⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj1033.3	⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
bnj1033.4	⊢ (𝜃 ↔ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴))
bnj1033.5	⊢ (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))
bnj1033.6	⊢ (𝜂 ↔ 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅))
bnj1033.7	⊢ (𝜁 ↔ (𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖)))
bnj1033.8	⊢ 𝐷 = (ω ∖ {∅})
bnj1033.9	⊢ 𝐾 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}
bnj1033.10	⊢ (∃𝑓∃𝑛∃𝑖(𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵)

Assertion

Ref	Expression
bnj1033	⊢ ((𝜃 ∧ 𝜏) → trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐵)

Distinct variable groups:   𝐴,𝑓,𝑖,𝑛,𝑦   𝑧,𝐴,𝑓,𝑖,𝑛   𝑧,𝐵   𝐷,𝑖   𝑅,𝑓,𝑖,𝑛,𝑦   𝑧,𝑅   𝑓,𝑋,𝑖,𝑛,𝑦   𝑧,𝑋   𝜏,𝑓,𝑖,𝑛,𝑧   𝜃,𝑓,𝑖,𝑛,𝑧   𝜑,𝑖

Allowed substitution hints:   𝜑(𝑦, 𝑧, 𝑓, 𝑛)   𝜓(𝑦, 𝑧, 𝑓, 𝑖, 𝑛)   𝜒(𝑦, 𝑧, 𝑓, 𝑖, 𝑛)   𝜃(𝑦)   𝜏(𝑦)   𝜂(𝑦, 𝑧, 𝑓, 𝑖, 𝑛)   𝜁(𝑦, 𝑧, 𝑓, 𝑖, 𝑛)   𝐵(𝑦, 𝑓, 𝑖, 𝑛)   𝐷(𝑦, 𝑧, 𝑓, 𝑛)   𝐾(𝑦, 𝑧, 𝑓, 𝑖, 𝑛)

Proof of Theorem bnj1033

Step	Hyp	Ref	Expression
1		bnj1033.1	. . . . 5 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
2		bnj1033.2	. . . . 5 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
3		bnj1033.8	. . . . 5 ⊢ 𝐷 = (ω ∖ {∅})
4		bnj1033.9	. . . . 5 ⊢ 𝐾 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}
5		bnj1033.3	. . . . 5 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
6	1, 2, 3, 4, 5	bnj983 35648	. . . 4 ⊢ (𝑧 ∈ trCl(𝑋, 𝐴, 𝑅) ↔ ∃𝑓∃𝑛∃𝑖(𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖)))
7		19.42v 1988	. . . . . . . . . 10 ⊢ (∃𝑖((𝜃 ∧ 𝜏) ∧ (𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖))) ↔ ((𝜃 ∧ 𝜏) ∧ ∃𝑖(𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖))))
8		df-3an 1105	. . . . . . . . . . 11 ⊢ ((𝜃 ∧ 𝜏 ∧ (𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖))) ↔ ((𝜃 ∧ 𝜏) ∧ (𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖))))
9	8	exbii 1881	. . . . . . . . . 10 ⊢ (∃𝑖(𝜃 ∧ 𝜏 ∧ (𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖))) ↔ ∃𝑖((𝜃 ∧ 𝜏) ∧ (𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖))))
10		df-3an 1105	. . . . . . . . . 10 ⊢ ((𝜃 ∧ 𝜏 ∧ ∃𝑖(𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖))) ↔ ((𝜃 ∧ 𝜏) ∧ ∃𝑖(𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖))))
11	7, 9, 10	3bitr4i 306	. . . . . . . . 9 ⊢ (∃𝑖(𝜃 ∧ 𝜏 ∧ (𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖))) ↔ (𝜃 ∧ 𝜏 ∧ ∃𝑖(𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖))))
12	11	exbii 1881	. . . . . . . 8 ⊢ (∃𝑛∃𝑖(𝜃 ∧ 𝜏 ∧ (𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖))) ↔ ∃𝑛(𝜃 ∧ 𝜏 ∧ ∃𝑖(𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖))))
13		19.42v 1988	. . . . . . . . 9 ⊢ (∃𝑛((𝜃 ∧ 𝜏) ∧ ∃𝑖(𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖))) ↔ ((𝜃 ∧ 𝜏) ∧ ∃𝑛∃𝑖(𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖))))
14	10	exbii 1881	. . . . . . . . 9 ⊢ (∃𝑛(𝜃 ∧ 𝜏 ∧ ∃𝑖(𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖))) ↔ ∃𝑛((𝜃 ∧ 𝜏) ∧ ∃𝑖(𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖))))
15		df-3an 1105	. . . . . . . . 9 ⊢ ((𝜃 ∧ 𝜏 ∧ ∃𝑛∃𝑖(𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖))) ↔ ((𝜃 ∧ 𝜏) ∧ ∃𝑛∃𝑖(𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖))))
16	13, 14, 15	3bitr4i 306	. . . . . . . 8 ⊢ (∃𝑛(𝜃 ∧ 𝜏 ∧ ∃𝑖(𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖))) ↔ (𝜃 ∧ 𝜏 ∧ ∃𝑛∃𝑖(𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖))))
17	12, 16	bitri 278	. . . . . . 7 ⊢ (∃𝑛∃𝑖(𝜃 ∧ 𝜏 ∧ (𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖))) ↔ (𝜃 ∧ 𝜏 ∧ ∃𝑛∃𝑖(𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖))))
18	17	exbii 1881	. . . . . 6 ⊢ (∃𝑓∃𝑛∃𝑖(𝜃 ∧ 𝜏 ∧ (𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖))) ↔ ∃𝑓(𝜃 ∧ 𝜏 ∧ ∃𝑛∃𝑖(𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖))))
19		19.42v 1988	. . . . . . 7 ⊢ (∃𝑓((𝜃 ∧ 𝜏) ∧ ∃𝑛∃𝑖(𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖))) ↔ ((𝜃 ∧ 𝜏) ∧ ∃𝑓∃𝑛∃𝑖(𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖))))
20	15	exbii 1881	. . . . . . 7 ⊢ (∃𝑓(𝜃 ∧ 𝜏 ∧ ∃𝑛∃𝑖(𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖))) ↔ ∃𝑓((𝜃 ∧ 𝜏) ∧ ∃𝑛∃𝑖(𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖))))
21		df-3an 1105	. . . . . . 7 ⊢ ((𝜃 ∧ 𝜏 ∧ ∃𝑓∃𝑛∃𝑖(𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖))) ↔ ((𝜃 ∧ 𝜏) ∧ ∃𝑓∃𝑛∃𝑖(𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖))))
22	19, 20, 21	3bitr4i 306	. . . . . 6 ⊢ (∃𝑓(𝜃 ∧ 𝜏 ∧ ∃𝑛∃𝑖(𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖))) ↔ (𝜃 ∧ 𝜏 ∧ ∃𝑓∃𝑛∃𝑖(𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖))))
23	18, 22	bitri 278	. . . . 5 ⊢ (∃𝑓∃𝑛∃𝑖(𝜃 ∧ 𝜏 ∧ (𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖))) ↔ (𝜃 ∧ 𝜏 ∧ ∃𝑓∃𝑛∃𝑖(𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖))))
24		bnj255 35403	. . . . . . . 8 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) ↔ (𝜃 ∧ 𝜏 ∧ (𝜒 ∧ 𝜁)))
25		bnj1033.7	. . . . . . . . . . 11 ⊢ (𝜁 ↔ (𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖)))
26	25	anbi2i 635	. . . . . . . . . 10 ⊢ ((𝜒 ∧ 𝜁) ↔ (𝜒 ∧ (𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖))))
27		3anass 1111	. . . . . . . . . 10 ⊢ ((𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖)) ↔ (𝜒 ∧ (𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖))))
28	26, 27	bitr4i 281	. . . . . . . . 9 ⊢ ((𝜒 ∧ 𝜁) ↔ (𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖)))
29	28	3anbi3i 1177	. . . . . . . 8 ⊢ ((𝜃 ∧ 𝜏 ∧ (𝜒 ∧ 𝜁)) ↔ (𝜃 ∧ 𝜏 ∧ (𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖))))
30	24, 29	bitri 278	. . . . . . 7 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) ↔ (𝜃 ∧ 𝜏 ∧ (𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖))))
31	30	3exbii 1883	. . . . . 6 ⊢ (∃𝑓∃𝑛∃𝑖(𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) ↔ ∃𝑓∃𝑛∃𝑖(𝜃 ∧ 𝜏 ∧ (𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖))))
32		bnj1033.10	. . . . . 6 ⊢ (∃𝑓∃𝑛∃𝑖(𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵)
33	31, 32	sylbir 238	. . . . 5 ⊢ (∃𝑓∃𝑛∃𝑖(𝜃 ∧ 𝜏 ∧ (𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖))) → 𝑧 ∈ 𝐵)
34	23, 33	sylbir 238	. . . 4 ⊢ ((𝜃 ∧ 𝜏 ∧ ∃𝑓∃𝑛∃𝑖(𝜒 ∧ 𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖))) → 𝑧 ∈ 𝐵)
35	6, 34	syl3an3b 1433	. . 3 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅)) → 𝑧 ∈ 𝐵)
36	35	3expia 1139	. 2 ⊢ ((𝜃 ∧ 𝜏) → (𝑧 ∈ trCl(𝑋, 𝐴, 𝑅) → 𝑧 ∈ 𝐵))
37	36	ssrdv 3919	1 ⊢ ((𝜃 ∧ 𝜏) → trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐵)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2147  {cab 2741  ∀wral 3080  ∃wrex 3090  Vcvv 3454   ⊆ wss 3900   ∖ cdif 4048  ∅c0 4281  {csn 4588  ∪ ciun 4954  suc csuc 6379   Fn wfn 6550  ‘cfv 6555  ωcom 7898   ∧ w-bnj17 35384   predc-bnj14 35386   FrSe w-bnj15 35390   trClc-bnj18 35392   TrFow-bnj19 35394

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-12 2216  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-v 3456  df-ss 3901  df-iun 4956  df-fn 6558  df-bnj17 35385  df-bnj18 35393

This theorem is used by:  bnj1034  35667