Theorem bnj907 35364

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
bnj907.1	⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj907.2	⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj907.3	⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
bnj907.4	⊢ (𝜃 ↔ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))
bnj907.5	⊢ (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛))
bnj907.6	⊢ (𝜂 ↔ (𝑖 ∈ 𝑛 ∧ 𝑦 ∈ (𝑓‘𝑖)))
bnj907.7	⊢ (𝜑′ ↔ [𝑝 / 𝑛]𝜑)
bnj907.8	⊢ (𝜓′ ↔ [𝑝 / 𝑛]𝜓)
bnj907.9	⊢ (𝜒′ ↔ [𝑝 / 𝑛]𝜒)
bnj907.10	⊢ (𝜑″ ↔ [𝐺 / 𝑓]𝜑′)
bnj907.11	⊢ (𝜓″ ↔ [𝐺 / 𝑓]𝜓′)
bnj907.12	⊢ (𝜒″ ↔ [𝐺 / 𝑓]𝜒′)
bnj907.13	⊢ 𝐷 = (ω ∖ {∅})
bnj907.14	⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}
bnj907.15	⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅)
bnj907.16	⊢ 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})

Assertion

Ref	Expression
bnj907	⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → TrFo( trCl(𝑋, 𝐴, 𝑅), 𝐴, 𝑅))

Distinct variable groups:   𝐴,𝑓,𝑖,𝑚,𝑛,𝑝,𝑦   𝑧,𝐴,𝑦   𝐷,𝑓,𝑖,𝑛   𝑖,𝐺,𝑝   𝑅,𝑓,𝑖,𝑚,𝑛,𝑝,𝑦   𝑧,𝑅   𝑓,𝑋,𝑖,𝑚,𝑛,𝑦   𝑧,𝑋   𝜒,𝑚,𝑝   𝜂,𝑚,𝑝   𝜃,𝑓,𝑖,𝑚,𝑛,𝑝   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, 𝑛)   𝜃(𝑦, 𝑧)   𝜏(𝑦, 𝑧, 𝑓, 𝑖, 𝑚, 𝑛, 𝑝)   𝜂(𝑦, 𝑧, 𝑓, 𝑖, 𝑛)   𝐵(𝑦, 𝑧, 𝑓, 𝑖, 𝑚, 𝑛, 𝑝)   𝐶(𝑦, 𝑧, 𝑓, 𝑖, 𝑚, 𝑛, 𝑝)   𝐷(𝑦, 𝑧, 𝑚, 𝑝)   𝐺(𝑦, 𝑧, 𝑓, 𝑚, 𝑛)   𝑋(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 bnj907

Step	Hyp	Ref	Expression
1		bnj907.4	. 2 ⊢ (𝜃 ↔ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))
2		bnj907.1	. . . . . . . . 9 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
3		bnj907.2	. . . . . . . . 9 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
4		bnj907.3	. . . . . . . . 9 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
5		bnj907.5	. . . . . . . . 9 ⊢ (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛))
6		bnj907.6	. . . . . . . . 9 ⊢ (𝜂 ↔ (𝑖 ∈ 𝑛 ∧ 𝑦 ∈ (𝑓‘𝑖)))
7		bnj907.13	. . . . . . . . 9 ⊢ 𝐷 = (ω ∖ {∅})
8		bnj907.14	. . . . . . . . 9 ⊢ 𝐵 = {𝑓 ∣ ∃𝑛 ∈ 𝐷 (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓)}
9	2, 3, 4, 1, 5, 6, 7, 8	bnj1021 35363	. . . . . . . 8 ⊢ ∃𝑓∃𝑛∃𝑖∃𝑚(𝜃 → (𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃𝑝𝜏))
10		bnj907.7	. . . . . . . . . . . 12 ⊢ (𝜑′ ↔ [𝑝 / 𝑛]𝜑)
11		bnj907.8	. . . . . . . . . . . 12 ⊢ (𝜓′ ↔ [𝑝 / 𝑛]𝜓)
12		bnj907.9	. . . . . . . . . . . 12 ⊢ (𝜒′ ↔ [𝑝 / 𝑛]𝜒)
13		bnj907.10	. . . . . . . . . . . 12 ⊢ (𝜑″ ↔ [𝐺 / 𝑓]𝜑′)
14		bnj907.11	. . . . . . . . . . . 12 ⊢ (𝜓″ ↔ [𝐺 / 𝑓]𝜓′)
15		bnj907.12	. . . . . . . . . . . 12 ⊢ (𝜒″ ↔ [𝐺 / 𝑓]𝜒′)
16		bnj907.15	. . . . . . . . . . . 12 ⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅)
17		bnj907.16	. . . . . . . . . . . 12 ⊢ 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
18		vex 3458	. . . . . . . . . . . . . 14 ⊢ 𝑝 ∈ V
19	4, 10, 11, 12, 18	bnj919 35165	. . . . . . . . . . . . 13 ⊢ (𝜒′ ↔ (𝑝 ∈ 𝐷 ∧ 𝑓 Fn 𝑝 ∧ 𝜑′ ∧ 𝜓′))
20	17	bnj918 35164	. . . . . . . . . . . . 13 ⊢ 𝐺 ∈ V
21	19, 13, 14, 15, 20	bnj976 35175	. . . . . . . . . . . 12 ⊢ (𝜒″ ↔ (𝑝 ∈ 𝐷 ∧ 𝐺 Fn 𝑝 ∧ 𝜑″ ∧ 𝜓″))
22	2, 3, 4, 1, 5, 6, 10, 11, 12, 13, 14, 15, 7, 8, 16, 17, 21	bnj1020 35362	. . . . . . . . . . 11 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃𝑝𝜏) → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
23	22	ax-gen 1824	. . . . . . . . . 10 ⊢ ∀𝑚((𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃𝑝𝜏) → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
24		19.29r 1903	. . . . . . . . . . 11 ⊢ ((∃𝑚(𝜃 → (𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃𝑝𝜏)) ∧ ∀𝑚((𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃𝑝𝜏) → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))) → ∃𝑚((𝜃 → (𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃𝑝𝜏)) ∧ ((𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃𝑝𝜏) → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))))
25		pm3.33 776	. . . . . . . . . . 11 ⊢ (((𝜃 → (𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃𝑝𝜏)) ∧ ((𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃𝑝𝜏) → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))) → (𝜃 → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)))
26	24, 25	bnj593 35143	. . . . . . . . . 10 ⊢ ((∃𝑚(𝜃 → (𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃𝑝𝜏)) ∧ ∀𝑚((𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃𝑝𝜏) → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))) → ∃𝑚(𝜃 → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)))
27	23, 26	mpan2 703	. . . . . . . . 9 ⊢ (∃𝑚(𝜃 → (𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃𝑝𝜏)) → ∃𝑚(𝜃 → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)))
28	27	2eximi 1865	. . . . . . . 8 ⊢ (∃𝑛∃𝑖∃𝑚(𝜃 → (𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃𝑝𝜏)) → ∃𝑛∃𝑖∃𝑚(𝜃 → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)))
29	9, 28	bnj101 35121	. . . . . . 7 ⊢ ∃𝑓∃𝑛∃𝑖∃𝑚(𝜃 → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
30		19.9v 2013	. . . . . . 7 ⊢ (∃𝑓∃𝑛∃𝑖∃𝑚(𝜃 → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)) ↔ ∃𝑛∃𝑖∃𝑚(𝜃 → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)))
31	29, 30	mpbi 233	. . . . . 6 ⊢ ∃𝑛∃𝑖∃𝑚(𝜃 → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
32		19.9v 2013	. . . . . 6 ⊢ (∃𝑛∃𝑖∃𝑚(𝜃 → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)) ↔ ∃𝑖∃𝑚(𝜃 → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)))
33	31, 32	mpbi 233	. . . . 5 ⊢ ∃𝑖∃𝑚(𝜃 → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
34		19.9v 2013	. . . . 5 ⊢ (∃𝑖∃𝑚(𝜃 → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)) ↔ ∃𝑚(𝜃 → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)))
35	33, 34	mpbi 233	. . . 4 ⊢ ∃𝑚(𝜃 → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
36		19.9v 2013	. . . 4 ⊢ (∃𝑚(𝜃 → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)) ↔ (𝜃 → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)))
37	35, 36	mpbi 233	. . 3 ⊢ (𝜃 → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
38	1	bnj1254 35206	. . 3 ⊢ (𝜃 → 𝑧 ∈ pred(𝑦, 𝐴, 𝑅))
39	37, 38	sseldd 3937	. 2 ⊢ (𝜃 → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅))
40	1, 39	bnj978 35346	1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → TrFo( trCl(𝑋, 𝐴, 𝑅), 𝐴, 𝑅))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1102  ∀wal 1567   = wceq 1569  ∃wex 1808   ∈ wcel 2142  {cab 2740  ∀wral 3078  ∃wrex 3088  [wsbc 3743   ∖ cdif 3901   ∪ cun 3902   ⊆ wss 3904  ∅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   TrFow-bnj19 35094

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  df-bnj18 35093  df-bnj19 35095

This theorem is used by:  bnj1029  35365