Theorem bnj998 35587

Description: Technical lemma for bnj69 35640. 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
bnj998.1	⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj998.2	⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj998.3	⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
bnj998.4	⊢ (𝜃 ↔ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))
bnj998.5	⊢ (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛))
bnj998.7	⊢ (𝜑′ ↔ [𝑝 / 𝑛]𝜑)
bnj998.8	⊢ (𝜓′ ↔ [𝑝 / 𝑛]𝜓)
bnj998.9	⊢ (𝜒′ ↔ [𝑝 / 𝑛]𝜒)
bnj998.10	⊢ (𝜑″ ↔ [𝐺 / 𝑓]𝜑′)
bnj998.11	⊢ (𝜓″ ↔ [𝐺 / 𝑓]𝜓′)
bnj998.12	⊢ (𝜒″ ↔ [𝐺 / 𝑓]𝜒′)
bnj998.13	⊢ 𝐷 = (ω ∖ {∅})
bnj998.14	⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}
bnj998.15	⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅)
bnj998.16	⊢ 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})

Assertion

Ref	Expression
bnj998	⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → 𝜒″)

Distinct variable groups:   𝐴,𝑓,𝑖,𝑚,𝑛,𝑦   𝐷,𝑓,𝑖,𝑛   𝑖,𝐺   𝑅,𝑓,𝑖,𝑚,𝑛,𝑦   𝑓,𝑋,𝑖,𝑛   𝑓,𝑝,𝑖,𝑛   𝜑,𝑖

Allowed substitution hints:   𝜑(𝑦, 𝑧, 𝑓, 𝑚, 𝑛, 𝑝)   𝜓(𝑦, 𝑧, 𝑓, 𝑖, 𝑚, 𝑛, 𝑝)   𝜒(𝑦, 𝑧, 𝑓, 𝑖, 𝑚, 𝑛, 𝑝)   𝜃(𝑦, 𝑧, 𝑓, 𝑖, 𝑚, 𝑛, 𝑝)   𝜏(𝑦, 𝑧, 𝑓, 𝑖, 𝑚, 𝑛, 𝑝)   𝜂(𝑦, 𝑧, 𝑓, 𝑖, 𝑚, 𝑛, 𝑝)   𝐴(𝑧, 𝑝)   𝐵(𝑦, 𝑧, 𝑓, 𝑖, 𝑚, 𝑛, 𝑝)   𝐶(𝑦, 𝑧, 𝑓, 𝑖, 𝑚, 𝑛, 𝑝)   𝐷(𝑦, 𝑧, 𝑚, 𝑝)   𝑅(𝑧, 𝑝)   𝐺(𝑦, 𝑧, 𝑓, 𝑚, 𝑛, 𝑝)   𝑋(𝑦, 𝑧, 𝑚, 𝑝)   𝜑′(𝑦, 𝑧, 𝑓, 𝑖, 𝑚, 𝑛, 𝑝)   𝜓′(𝑦, 𝑧, 𝑓, 𝑖, 𝑚, 𝑛, 𝑝)   𝜒′(𝑦, 𝑧, 𝑓, 𝑖, 𝑚, 𝑛, 𝑝)   𝜑″(𝑦, 𝑧, 𝑓, 𝑖, 𝑚, 𝑛, 𝑝)   𝜓″(𝑦, 𝑧, 𝑓, 𝑖, 𝑚, 𝑛, 𝑝)   𝜒″(𝑦, 𝑧, 𝑓, 𝑖, 𝑚, 𝑛, 𝑝)

Proof of Theorem bnj998

Step	Hyp	Ref	Expression
1		bnj998.4	. . . . . 6 ⊢ (𝜃 ↔ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))
2		bnj253 35335	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))
3	2	simp1bi 1163	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)) → (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴))
4	1, 3	sylbi 220	. . . . 5 ⊢ (𝜃 → (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴))
5	4	bnj705 35384	. . . 4 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴))
6		bnj643 35380	. . . 4 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → 𝜒)
7		bnj998.5	. . . . . 6 ⊢ (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛))
8		3simpc 1168	. . . . . 6 ⊢ ((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) → (𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛))
9	7, 8	sylbi 220	. . . . 5 ⊢ (𝜏 → (𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛))
10	9	bnj707 35386	. . . 4 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → (𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛))
11		bnj255 35336	. . . 4 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝜒 ∧ (𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)))
12	5, 6, 10, 11	syl3anbrc 1362	. . 3 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛))
13		bnj252 35334	. . 3 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ↔ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)))
14	12, 13	sylib 221	. 2 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)))
15		bnj998.1	. . 3 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
16		bnj998.2	. . 3 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
17		bnj998.3	. . 3 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
18		bnj998.7	. . 3 ⊢ (𝜑′ ↔ [𝑝 / 𝑛]𝜑)
19		bnj998.8	. . 3 ⊢ (𝜓′ ↔ [𝑝 / 𝑛]𝜓)
20		bnj998.9	. . 3 ⊢ (𝜒′ ↔ [𝑝 / 𝑛]𝜒)
21		bnj998.10	. . 3 ⊢ (𝜑″ ↔ [𝐺 / 𝑓]𝜑′)
22		bnj998.11	. . 3 ⊢ (𝜓″ ↔ [𝐺 / 𝑓]𝜓′)
23		bnj998.12	. . 3 ⊢ (𝜒″ ↔ [𝐺 / 𝑓]𝜒′)
24		bnj998.13	. . 3 ⊢ 𝐷 = (ω ∖ {∅})
25		bnj998.14	. . 3 ⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}
26		bnj998.15	. . 3 ⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅)
27		bnj998.16	. . 3 ⊢ 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
28		biid 264	. . 3 ⊢ ((𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
29		biid 264	. . 3 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑚 ∈ 𝑛) ↔ (𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑚 ∈ 𝑛))
30	15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29	bnj910 35578	. 2 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝜒″)
31	14, 30	syl 18	1 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → 𝜒″)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  {cab 2738  ∀wral 3076  ∃wrex 3086  [wsbc 3738   ∖ cdif 4045   ∪ cun 4149  ∅c0 4278  {csn 4583  ⟨cop 4589  ∪ ciun 4950  suc csuc 6364   Fn wfn 6533  ‘cfv 6538  ωcom 7877   ∧ w-bnj17 35317   predc-bnj14 35319   FrSe w-bnj15 35323   trClc-bnj18 35325

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7751  ax-reg 9586

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  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 3739  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-res 5662  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-fv 6546  df-om 7878  df-bnj17 35318  df-bnj14 35320  df-bnj13 35322  df-bnj15 35324

This theorem is used by:  bnj1020  35595