Theorem bnj967 35107

Description: Technical lemma for bnj69 35172. 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
bnj967.2	⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj967.3	⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
bnj967.10	⊢ 𝐷 = (ω ∖ {∅})
bnj967.12	⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅)
bnj967.13	⊢ 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
bnj967.44	⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝐶 ∈ V)

Assertion

Ref	Expression
bnj967	⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ suc 𝑖 ∈ 𝑛)) → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅))

Distinct variable groups:   𝑦,𝑓   𝑦,𝑖   𝑦,𝑛

Allowed substitution hints:   𝜑(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜓(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜒(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐴(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐶(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐷(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝑅(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐺(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝑋(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)

Proof of Theorem bnj967

Step	Hyp	Ref	Expression
1		bnj967.44	. . . . . . 7 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛)) → 𝐶 ∈ V)
2	1	3adant3 1133	. . . . . 6 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ suc 𝑖 ∈ 𝑛)) → 𝐶 ∈ V)
3		bnj967.3	. . . . . . . . 9 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
4	3	bnj1235 34966	. . . . . . . 8 ⊢ (𝜒 → 𝑓 Fn 𝑛)
5	4	3ad2ant1 1134	. . . . . . 7 ⊢ ((𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) → 𝑓 Fn 𝑛)
6	5	3ad2ant2 1135	. . . . . 6 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ suc 𝑖 ∈ 𝑛)) → 𝑓 Fn 𝑛)
7		simp23 1210	. . . . . 6 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ suc 𝑖 ∈ 𝑛)) → 𝑝 = suc 𝑛)
8		simp3 1139	. . . . . . 7 ⊢ ((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ suc 𝑖 ∈ 𝑛) → suc 𝑖 ∈ 𝑛)
9	8	3ad2ant3 1136	. . . . . 6 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ suc 𝑖 ∈ 𝑛)) → suc 𝑖 ∈ 𝑛)
10	2, 6, 7, 9	bnj951 34938	. . . . 5 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ suc 𝑖 ∈ 𝑛)) → (𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛 ∧ suc 𝑖 ∈ 𝑛))
11		bnj967.10	. . . . . . . . . 10 ⊢ 𝐷 = (ω ∖ {∅})
12	11	bnj923 34931	. . . . . . . . 9 ⊢ (𝑛 ∈ 𝐷 → 𝑛 ∈ ω)
13	3, 12	bnj769 34925	. . . . . . . 8 ⊢ (𝜒 → 𝑛 ∈ ω)
14	13	3ad2ant1 1134	. . . . . . 7 ⊢ ((𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) → 𝑛 ∈ ω)
15	14, 8	bnj240 34862	. . . . . 6 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ suc 𝑖 ∈ 𝑛)) → (𝑛 ∈ ω ∧ suc 𝑖 ∈ 𝑛))
16		nnord 7820	. . . . . . . 8 ⊢ (𝑛 ∈ ω → Ord 𝑛)
17		ordtr 6333	. . . . . . . 8 ⊢ (Ord 𝑛 → Tr 𝑛)
18	16, 17	syl 17	. . . . . . 7 ⊢ (𝑛 ∈ ω → Tr 𝑛)
19		trsuc 6408	. . . . . . 7 ⊢ ((Tr 𝑛 ∧ suc 𝑖 ∈ 𝑛) → 𝑖 ∈ 𝑛)
20	18, 19	sylan 581	. . . . . 6 ⊢ ((𝑛 ∈ ω ∧ suc 𝑖 ∈ 𝑛) → 𝑖 ∈ 𝑛)
21	15, 20	syl 17	. . . . 5 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ suc 𝑖 ∈ 𝑛)) → 𝑖 ∈ 𝑛)
22		bnj658 34914	. . . . . . 7 ⊢ ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛 ∧ suc 𝑖 ∈ 𝑛) → (𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛))
23	22	anim1i 616	. . . . . 6 ⊢ (((𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛 ∧ suc 𝑖 ∈ 𝑛) ∧ 𝑖 ∈ 𝑛) → ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛) ∧ 𝑖 ∈ 𝑛))
24		df-bnj17 34850	. . . . . 6 ⊢ ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛 ∧ 𝑖 ∈ 𝑛) ↔ ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛) ∧ 𝑖 ∈ 𝑛))
25	23, 24	sylibr 234	. . . . 5 ⊢ (((𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛 ∧ suc 𝑖 ∈ 𝑛) ∧ 𝑖 ∈ 𝑛) → (𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛 ∧ 𝑖 ∈ 𝑛))
26	10, 21, 25	syl2anc 585	. . . 4 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ suc 𝑖 ∈ 𝑛)) → (𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛 ∧ 𝑖 ∈ 𝑛))
27		bnj967.13	. . . . 5 ⊢ 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
28	27	bnj945 34936	. . . 4 ⊢ ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛 ∧ 𝑖 ∈ 𝑛) → (𝐺‘𝑖) = (𝑓‘𝑖))
29	26, 28	syl 17	. . 3 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ suc 𝑖 ∈ 𝑛)) → (𝐺‘𝑖) = (𝑓‘𝑖))
30	27	bnj945 34936	. . . 4 ⊢ ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛 ∧ suc 𝑖 ∈ 𝑛) → (𝐺‘suc 𝑖) = (𝑓‘suc 𝑖))
31	10, 30	syl 17	. . 3 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ suc 𝑖 ∈ 𝑛)) → (𝐺‘suc 𝑖) = (𝑓‘suc 𝑖))
32		3simpb 1150	. . . 4 ⊢ ((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ suc 𝑖 ∈ 𝑛) → (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛))
33	32	3ad2ant3 1136	. . 3 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ suc 𝑖 ∈ 𝑛)) → (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛))
34	3	bnj1254 34971	. . . . 5 ⊢ (𝜒 → 𝜓)
35	34	3ad2ant1 1134	. . . 4 ⊢ ((𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) → 𝜓)
36	35	3ad2ant2 1135	. . 3 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ suc 𝑖 ∈ 𝑛)) → 𝜓)
37	29, 31, 33, 36	bnj951 34938	. 2 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ suc 𝑖 ∈ 𝑛)) → ((𝐺‘𝑖) = (𝑓‘𝑖) ∧ (𝐺‘suc 𝑖) = (𝑓‘suc 𝑖) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛) ∧ 𝜓))
38		bnj967.2	. . 3 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑛 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
39		bnj967.12	. . . 4 ⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑚) pred(𝑦, 𝐴, 𝑅)
40	39, 27	bnj958 35102	. . 3 ⊢ ((𝐺‘𝑖) = (𝑓‘𝑖) → ∀𝑦(𝐺‘𝑖) = (𝑓‘𝑖))
41	38, 40	bnj953 35101	. 2 ⊢ (((𝐺‘𝑖) = (𝑓‘𝑖) ∧ (𝐺‘suc 𝑖) = (𝑓‘suc 𝑖) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛) ∧ 𝜓) → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅))
42	37, 41	syl 17	1 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝑋 ∈ 𝐴) ∧ (𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ suc 𝑖 ∈ 𝑛)) → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∀wral 3052  Vcvv 3430   ∖ cdif 3887   ∪ cun 3888  ∅c0 4274  {csn 4568  ⟨cop 4574  ∪ ciun 4934  Tr wtr 5193  Ord word 6318  suc csuc 6321   Fn wfn 6489  ‘cfv 6494  ωcom 7812   ∧ w-bnj17 34849   predc-bnj14 34851   FrSe w-bnj15 34855

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pr 5372  ax-reg 9502

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-tr 5194  df-id 5521  df-po 5534  df-so 5535  df-fr 5579  df-we 5581  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-res 5638  df-ord 6322  df-on 6323  df-suc 6325  df-iota 6450  df-fun 6496  df-fn 6497  df-fv 6502  df-om 7813  df-bnj17 34850

This theorem is referenced by:  bnj910  35110