Theorem bnj967 35142

Description: Technical lemma for bnj69 35207. 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 1139	. . . . . 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 35001	. . . . . . . 8 ⊢ (𝜒 → 𝑓 Fn 𝑛)
5	4	3ad2ant1 1140	. . . . . . 7 ⊢ ((𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) → 𝑓 Fn 𝑛)
6	5	3ad2ant2 1141	. . . . . 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 1216	. . . . . 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 1145	. . . . . . 7 ⊢ ((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ suc 𝑖 ∈ 𝑛) → suc 𝑖 ∈ 𝑛)
9	8	3ad2ant3 1142	. . . . . 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 34973	. . . . 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 34966	. . . . . . . . 9 ⊢ (𝑛 ∈ 𝐷 → 𝑛 ∈ ω)
13	3, 12	bnj769 34960	. . . . . . . 8 ⊢ (𝜒 → 𝑛 ∈ ω)
14	13	3ad2ant1 1140	. . . . . . 7 ⊢ ((𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) → 𝑛 ∈ ω)
15	14, 8	bnj240 34897	. . . . . 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 7818	. . . . . . . 8 ⊢ (𝑛 ∈ ω → Ord 𝑛)
17		ordtr 6328	. . . . . . . 8 ⊢ (Ord 𝑛 → Tr 𝑛)
18	16, 17	syl 17	. . . . . . 7 ⊢ (𝑛 ∈ ω → Tr 𝑛)
19		trsuc 6403	. . . . . . 7 ⊢ ((Tr 𝑛 ∧ suc 𝑖 ∈ 𝑛) → 𝑖 ∈ 𝑛)
20	18, 19	sylan 587	. . . . . 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 34949	. . . . . . 7 ⊢ ((𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛 ∧ suc 𝑖 ∈ 𝑛) → (𝐶 ∈ V ∧ 𝑓 Fn 𝑛 ∧ 𝑝 = suc 𝑛))
23	22	anim1i 622	. . . . . 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 34885	. . . . . 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 236	. . . . 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 591	. . . 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 34971	. . . 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 34971	. . . 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 1156	. . . 4 ⊢ ((𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑝 ∧ suc 𝑖 ∈ 𝑛) → (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛))
33	32	3ad2ant3 1142	. . 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 35006	. . . . 5 ⊢ (𝜒 → 𝜓)
35	34	3ad2ant1 1140	. . . 4 ⊢ ((𝜒 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 = suc 𝑛) → 𝜓)
36	35	3ad2ant2 1141	. . 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 34973	. 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 35137	. . 3 ⊢ ((𝐺‘𝑖) = (𝑓‘𝑖) → ∀𝑦(𝐺‘𝑖) = (𝑓‘𝑖))
41	38, 40	bnj953 35136	. 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 208   ∧ wa 397   ∧ w3a 1093   = wceq 1548   ∈ wcel 2121  ∀wral 3055  Vcvv 3433   ∖ cdif 3882   ∪ cun 3883  ∅c0 4264  {csn 4558  ⟨cop 4564  ∪ ciun 4924  Tr wtr 5182  Ord word 6313  suc csuc 6316   Fn wfn 6484  ‘cfv 6489  ωcom 7810   ∧ w-bnj17 34884   predc-bnj14 34886   FrSe w-bnj15 34890

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-nul 5231  ax-pr 5365  ax-reg 9501

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-iun 4926  df-br 5076  df-opab 5138  df-tr 5183  df-id 5516  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-res 5633  df-ord 6317  df-on 6318  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-fv 6497  df-om 7811  df-bnj17 34885

This theorem is referenced by:  bnj910  35145