Theorem bnj1053 35526

Description: Technical lemma for bnj69 35560. 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
bnj1053.1	⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj1053.2	⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj1053.3	⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
bnj1053.4	⊢ (𝜃 ↔ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴))
bnj1053.5	⊢ (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))
bnj1053.6	⊢ (𝜁 ↔ (𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖)))
bnj1053.7	⊢ 𝐷 = (ω ∖ {∅})
bnj1053.8	⊢ 𝐾 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}
bnj1053.9	⊢ (𝜂 ↔ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵))
bnj1053.10	⊢ (𝜌 ↔ ∀𝑗 ∈ 𝑛 (𝑗 E 𝑖 → [𝑗 / 𝑖]𝜂))
bnj1053.37	⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → ∀𝑖 ∈ 𝑛 (𝜌 → 𝜂))

Assertion

Ref	Expression
bnj1053	⊢ ((𝜃 ∧ 𝜏) → trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐵)

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

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

Proof of Theorem bnj1053

Step	Hyp	Ref	Expression
1		bnj1053.1	. 2 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
2		bnj1053.2	. 2 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
3		bnj1053.3	. 2 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
4		bnj1053.4	. 2 ⊢ (𝜃 ↔ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴))
5		bnj1053.5	. 2 ⊢ (𝜏 ↔ (𝐵 ∈ V ∧ TrFo(𝐵, 𝐴, 𝑅) ∧ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐵))
6		bnj1053.6	. 2 ⊢ (𝜁 ↔ (𝑖 ∈ 𝑛 ∧ 𝑧 ∈ (𝑓‘𝑖)))
7		bnj1053.7	. 2 ⊢ 𝐷 = (ω ∖ {∅})
8		bnj1053.8	. 2 ⊢ 𝐾 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}
9		bnj1053.9	. 2 ⊢ (𝜂 ↔ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → 𝑧 ∈ 𝐵))
10		bnj1053.10	. 2 ⊢ (𝜌 ↔ ∀𝑗 ∈ 𝑛 (𝑗 E 𝑖 → [𝑗 / 𝑖]𝜂))
11	7	bnj923 35319	. . . . . 6 ⊢ (𝑛 ∈ 𝐷 → 𝑛 ∈ ω)
12		nnord 7869	. . . . . 6 ⊢ (𝑛 ∈ ω → Ord 𝑛)
13		ordfr 6367	. . . . . 6 ⊢ (Ord 𝑛 → E Fr 𝑛)
14	11, 12, 13	3syl 19	. . . . 5 ⊢ (𝑛 ∈ 𝐷 → E Fr 𝑛)
15	3, 14	bnj769 35313	. . . 4 ⊢ (𝜒 → E Fr 𝑛)
16	15	bnj707 35306	. . 3 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → E Fr 𝑛)
17		bnj1053.37	. . 3 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → ∀𝑖 ∈ 𝑛 (𝜌 → 𝜂))
18	16, 17	jca 521	. 2 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒 ∧ 𝜁) → ( E Fr 𝑛 ∧ ∀𝑖 ∈ 𝑛 (𝜌 → 𝜂)))
19	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 18	bnj1052 35525	1 ⊢ ((𝜃 ∧ 𝜏) → trCl(𝑋, 𝐴, 𝑅) ⊆ 𝐵)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  {cab 2738  ∀wral 3076  ∃wrex 3086  Vcvv 3450  [wsbc 3739   ∖ cdif 3896   ⊆ wss 3899  ∅c0 4279  {csn 4584  ∪ ciun 4951   class class class wbr 5103   E cep 5547   Fr wfr 5598  Ord word 6351  suc csuc 6354   Fn wfn 6523  ‘cfv 6528  ωcom 7861   ∧ w-bnj17 35237   predc-bnj14 35239   FrSe w-bnj15 35243   trClc-bnj18 35245   TrFow-bnj19 35247

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5249

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-tr 5213  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-ord 6355  df-on 6356  df-fn 6531  df-om 7862  df-bnj17 35238  df-bnj18 35246

This theorem is used by: (None)