Theorem bnj907 35517

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
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 35516	. . . . . . . 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 3454	. . . . . . . . . . . . . 14 ⊢ 𝑝 ∈ V
19	4, 10, 11, 12, 18	bnj919 35318	. . . . . . . . . . . . 13 ⊢ (𝜒′ ↔ (𝑝 ∈ 𝐷 ∧ 𝑓 Fn 𝑝 ∧ 𝜑′ ∧ 𝜓′))
20	17	bnj918 35317	. . . . . . . . . . . . 13 ⊢ 𝐺 ∈ V
21	19, 13, 14, 15, 20	bnj976 35328	. . . . . . . . . . . 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 35515	. . . . . . . . . . 11 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃𝑝𝜏) → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
23	22	ax-gen 1828	. . . . . . . . . 10 ⊢ ∀𝑚((𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃𝑝𝜏) → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
24		19.29r 1907	. . . . . . . . . . 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		pm3.33 777	. . . . . . . . . . 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 35296	. . . . . . . . . 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 704	. . . . . . . . 9 ⊢ (∃𝑚(𝜃 → (𝜃 ∧ 𝜒 ∧ 𝜂 ∧ ∃𝑝𝜏)) → ∃𝑚(𝜃 → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅)))
28	27	2eximi 1869	. . . . . . . 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 35274	. . . . . . 7 ⊢ ∃𝑓∃𝑛∃𝑖∃𝑚(𝜃 → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
30		19.9v 2017	. . . . . . 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 2017	. . . . . 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 2017	. . . . 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 2017	. . . 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 35359	. . 3 ⊢ (𝜃 → 𝑧 ∈ pred(𝑦, 𝐴, 𝑅))
39	37, 38	sseldd 3932	. 2 ⊢ (𝜃 → 𝑧 ∈ trCl(𝑋, 𝐴, 𝑅))
40	1, 39	bnj978 35499	1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) → TrFo( trCl(𝑋, 𝐴, 𝑅), 𝐴, 𝑅))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2145  {cab 2738  ∀wral 3076  ∃wrex 3086  [wsbc 3739   ∖ cdif 3896   ∪ cun 3897   ⊆ wss 3899  ∅c0 4279  {csn 4584  ⟨cop 4590  ∪ ciun 4951  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-rep 5232  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7735  ax-reg 9564

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  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 3740  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  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-opab 5168  df-tr 5213  df-id 5543  df-eprel 5548  df-po 5556  df-so 5557  df-fr 5601  df-we 5603  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-res 5660  df-ord 6355  df-on 6356  df-lim 6357  df-suc 6358  df-iota 6484  df-fun 6530  df-fn 6531  df-fv 6536  df-om 7862  df-bnj17 35238  df-bnj14 35240  df-bnj13 35242  df-bnj15 35244  df-bnj18 35246  df-bnj19 35248

This theorem is used by:  bnj1029  35518