Theorem bnj1006 32461

Description: Technical lemma for bnj69 32511. 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
bnj1006.1	⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj1006.2	⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj1006.3	⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
bnj1006.4	⊢ (𝜃 ↔ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))
bnj1006.5	⊢ (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛))
bnj1006.6	⊢ (𝜂 ↔ (𝑖 ∈ 𝑛 ∧ 𝑦 ∈ (𝑓‘𝑖)))
bnj1006.7	⊢ (𝜑′ ↔ [𝑝 / 𝑛]𝜑)
bnj1006.8	⊢ (𝜓′ ↔ [𝑝 / 𝑛]𝜓)
bnj1006.9	⊢ (𝜒′ ↔ [𝑝 / 𝑛]𝜒)
bnj1006.10	⊢ (𝜑″ ↔ [𝐺 / 𝑓]𝜑′)
bnj1006.11	⊢ (𝜓″ ↔ [𝐺 / 𝑓]𝜓′)
bnj1006.12	⊢ (𝜒″ ↔ [𝐺 / 𝑓]𝜒′)
bnj1006.13	⊢ 𝐷 = (ω ∖ {∅})
bnj1006.15	⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅)
bnj1006.16	⊢ 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
bnj1006.28	⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → (𝜒″ ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝))

Assertion

Ref	Expression
bnj1006	⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → pred(𝑦, 𝐴, 𝑅) ⊆ (𝐺‘suc 𝑖))

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

Allowed substitution hints:   𝜑(𝑦,𝑧,𝑓,𝑖,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)   𝐶(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐷(𝑦,𝑧,𝑖,𝑚,𝑝)   𝑅(𝑧,𝑝)   𝐺(𝑦,𝑧,𝑓,𝑚,𝑛,𝑝)   𝑋(𝑦,𝑧,𝑖,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 bnj1006

Step	Hyp	Ref	Expression
1		bnj1006.6	. . . . 5 ⊢ (𝜂 ↔ (𝑖 ∈ 𝑛 ∧ 𝑦 ∈ (𝑓‘𝑖)))
2	1	simprbi 501	. . . 4 ⊢ (𝜂 → 𝑦 ∈ (𝑓‘𝑖))
3	2	bnj708 32256	. . 3 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → 𝑦 ∈ (𝑓‘𝑖))
4		bnj1006.4	. . . . . . . 8 ⊢ (𝜃 ↔ (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))
5		bnj253 32203	. . . . . . . . 9 ⊢ ((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)))
6	5	simp1bi 1143	. . . . . . . 8 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)) → (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴))
7	4, 6	sylbi 220	. . . . . . 7 ⊢ (𝜃 → (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴))
8	7	bnj705 32253	. . . . . 6 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → (𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴))
9		bnj643 32249	. . . . . . 7 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → 𝜒)
10		bnj1006.5	. . . . . . . . 9 ⊢ (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛))
11		3simpc 1148	. . . . . . . . 9 ⊢ ((𝑚 ∈ ω ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) → (𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛))
12	10, 11	sylbi 220	. . . . . . . 8 ⊢ (𝜏 → (𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛))
13	12	bnj707 32255	. . . . . . 7 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → (𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛))
14		3anass 1093	. . . . . . 7 ⊢ ((𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ↔ (𝜒 ∧ (𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)))
15	9, 13, 14	sylanbrc 587	. . . . . 6 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛))
16		bnj1006.1	. . . . . . 7 ⊢ (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
17		bnj1006.2	. . . . . . 7 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
18		bnj1006.3	. . . . . . 7 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
19		bnj1006.13	. . . . . . 7 ⊢ 𝐷 = (ω ∖ {∅})
20		bnj1006.15	. . . . . . 7 ⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅)
21		biid 264	. . . . . . 7 ⊢ ((𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓) ↔ (𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
22		biid 264	. . . . . . 7 ⊢ ((𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑚 ∈ 𝑛) ↔ (𝑛 ∈ 𝐷 ∧ 𝑝 = suc 𝑛 ∧ 𝑚 ∈ 𝑛))
23	16, 17, 18, 19, 20, 21, 22	bnj969 32447	. . . . . 6 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝐶 ∈ V)
24	8, 15, 23	syl2anc 588	. . . . 5 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → 𝐶 ∈ V)
25	18	bnj1235 32305	. . . . . 6 ⊢ (𝜒 → 𝑓 Fn 𝑛)
26	25	bnj706 32254	. . . . 5 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → 𝑓 Fn 𝑛)
27	10	simp3bi 1145	. . . . . 6 ⊢ (𝜏 → 𝑝 = suc 𝑛)
28	27	bnj707 32255	. . . . 5 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → 𝑝 = suc 𝑛)
29	1	simplbi 502	. . . . . 6 ⊢ (𝜂 → 𝑖 ∈ 𝑛)
30	29	bnj708 32256	. . . . 5 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → 𝑖 ∈ 𝑛)
31	24, 26, 28, 30	bnj951 32276	. . . 4 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → (𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛 ∧ 𝑖 ∈ 𝑛))
32		bnj1006.16	. . . . 5 ⊢ 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
33	32	bnj945 32274	. . . 4 ⊢ ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛 ∧ 𝑖 ∈ 𝑛) → (𝐺‘𝑖) = (𝑓‘𝑖))
34	31, 33	syl 17	. . 3 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → (𝐺‘𝑖) = (𝑓‘𝑖))
35	3, 34	eleqtrrd 2856	. 2 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → 𝑦 ∈ (𝐺‘𝑖))
36		bnj1006.28	. . . . 5 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → (𝜒″ ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝))
37	36	anim1i 618	. . . 4 ⊢ (((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) ∧ 𝑦 ∈ (𝐺‘𝑖)) → ((𝜒″ ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝) ∧ 𝑦 ∈ (𝐺‘𝑖)))
38		df-bnj17 32186	. . . 4 ⊢ ((𝜒″ ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑦 ∈ (𝐺‘𝑖)) ↔ ((𝜒″ ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝) ∧ 𝑦 ∈ (𝐺‘𝑖)))
39	37, 38	sylibr 237	. . 3 ⊢ (((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) ∧ 𝑦 ∈ (𝐺‘𝑖)) → (𝜒″ ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑦 ∈ (𝐺‘𝑖)))
40		bnj1006.7	. . . 4 ⊢ (𝜑′ ↔ [𝑝 / 𝑛]𝜑)
41		bnj1006.8	. . . 4 ⊢ (𝜓′ ↔ [𝑝 / 𝑛]𝜓)
42		bnj1006.9	. . . 4 ⊢ (𝜒′ ↔ [𝑝 / 𝑛]𝜒)
43		bnj1006.10	. . . 4 ⊢ (𝜑″ ↔ [𝐺 / 𝑓]𝜑′)
44		bnj1006.11	. . . 4 ⊢ (𝜓″ ↔ [𝐺 / 𝑓]𝜓′)
45		bnj1006.12	. . . 4 ⊢ (𝜒″ ↔ [𝐺 / 𝑓]𝜒′)
46	16, 17, 18, 40, 41, 42, 43, 44, 45, 20, 32	bnj999 32459	. . 3 ⊢ ((𝜒″ ∧ 𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ 𝑦 ∈ (𝐺‘𝑖)) → pred(𝑦, 𝐴, 𝑅) ⊆ (𝐺‘suc 𝑖))
47	39, 46	syl 17	. 2 ⊢ (((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) ∧ 𝑦 ∈ (𝐺‘𝑖)) → pred(𝑦, 𝐴, 𝑅) ⊆ (𝐺‘suc 𝑖))
48	35, 47	mpdan 687	1 ⊢ ((𝜃 ∧ 𝜒 ∧ 𝜏 ∧ 𝜂) → pred(𝑦, 𝐴, 𝑅) ⊆ (𝐺‘suc 𝑖))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1085   = wceq 1539   ∈ wcel 2112  ∀wral 3071  Vcvv 3410  [wsbc 3697   ∖ cdif 3856   ∪ cun 3857   ⊆ wss 3859  ∅c0 4226  {csn 4523  ⟨cop 4529  ∪ ciun 4884  suc csuc 6172   Fn wfn 6331  ‘cfv 6336  ωcom 7580   ∧ w-bnj17 32185   predc-bnj14 32187   FrSe w-bnj15 32191   trClc-bnj18 32193

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730  ax-rep 5157  ax-sep 5170  ax-nul 5177  ax-pr 5299  ax-un 7460  ax-reg 9090

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-mo 2558  df-eu 2589  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-ral 3076  df-rex 3077  df-reu 3078  df-rab 3080  df-v 3412  df-sbc 3698  df-csb 3807  df-dif 3862  df-un 3864  df-in 3866  df-ss 3876  df-pss 3878  df-nul 4227  df-if 4422  df-pw 4497  df-sn 4524  df-pr 4526  df-tp 4528  df-op 4530  df-uni 4800  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5431  df-eprel 5436  df-po 5444  df-so 5445  df-fr 5484  df-we 5486  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-ord 6173  df-on 6174  df-lim 6175  df-suc 6176  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-om 7581  df-bnj17 32186  df-bnj14 32188  df-bnj13 32190  df-bnj15 32192

This theorem is referenced by:  bnj1020  32466