Theorem bnj998 35354

Description: Technical lemma for bnj69 35407. 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 35102	. . . . . . 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 1162	. . . . . 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 35151	. . . 4 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴))
6		bnj643 35147	. . . 4 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → 𝜒)
7		bnj998.5	. . . . . 6 ⊢ (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛))
8		3simpc 1167	. . . . . 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 35153	. . . 4 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → (𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛))
11		bnj255 35103	. . . 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 1361	. . 3 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ 𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛))
13		bnj252 35101	. . 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 35345	. 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 400   ∧ w3a 1102   = wceq 1569   ∈ wcel 2142  {cab 2740  ∀wral 3078  ∃wrex 3088  [wsbc 3743   ∖ cdif 3901   ∪ cun 3902  ∅c0 4285  {csn 4588  ⟨cop 4594  ∪ ciun 4955  suc csuc 6362   Fn wfn 6531  ‘cfv 6536  ωcom 7860   ∧ w-bnj17 35084   predc-bnj14 35086   FrSe w-bnj15 35090   trClc-bnj18 35092

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-rep 5237  ax-sep 5256  ax-nul 5268  ax-pr 5403  ax-un 7734  ax-reg 9552

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1103  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-sbc 3744  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-pss 3924  df-nul 4286  df-if 4487  df-pw 4563  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-iun 4957  df-br 5109  df-opab 5173  df-tr 5218  df-id 5555  df-eprel 5560  df-po 5568  df-so 5569  df-fr 5613  df-we 5615  df-xp 5666  df-rel 5667  df-cnv 5668  df-co 5669  df-dm 5670  df-res 5672  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-fv 6544  df-om 7861  df-bnj17 35085  df-bnj14 35087  df-bnj13 35089  df-bnj15 35091

This theorem is used by:  bnj1020  35362