Theorem bnj570 35602

Description: Technical lemma for bnj852 35618. 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
bnj570.3	⊢ 𝐷 = (ω ∖ {∅})
bnj570.17	⊢ (𝜏 ↔ (𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′))
bnj570.19	⊢ (𝜂 ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
bnj570.21	⊢ (𝜌 ↔ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛 ∧ 𝑚 ≠ suc 𝑖))
bnj570.24	⊢ 𝐾 = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)
bnj570.26	⊢ 𝐺 = (𝑓 ∪ {⟨𝑚, 𝐶⟩})
bnj570.40	⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) → 𝐺 Fn 𝑛)
bnj570.30	⊢ (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑚 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))

Assertion

Ref	Expression
bnj570	⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜌) → (𝐺‘suc 𝑖) = 𝐾)

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

Allowed substitution hints:   𝜏(𝑦, 𝑓, 𝑖, 𝑚, 𝑛, 𝑝)   𝜂(𝑦, 𝑓, 𝑖, 𝑚, 𝑛, 𝑝)   𝜌(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(𝑓, 𝑖, 𝑚, 𝑛, 𝑝)   𝐾(𝑦, 𝑓, 𝑖, 𝑚, 𝑛, 𝑝)   𝜑′(𝑦, 𝑓, 𝑖, 𝑚, 𝑛, 𝑝)   𝜓′(𝑦, 𝑓, 𝑖, 𝑚, 𝑛, 𝑝)

Proof of Theorem bnj570

Step	Hyp	Ref	Expression
1		bnj251 35400	. . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜌) ↔ (𝑅 FrSe 𝐴 ∧ (𝜏 ∧ (𝜂 ∧ 𝜌))))
2		bnj570.17	. . . . . 6 ⊢ (𝜏 ↔ (𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′))
3	2	simp3bi 1165	. . . . 5 ⊢ (𝜏 → 𝜓′)
4		bnj570.21	. . . . . . . 8 ⊢ (𝜌 ↔ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛 ∧ 𝑚 ≠ suc 𝑖))
5	4	simp1bi 1163	. . . . . . 7 ⊢ (𝜌 → 𝑖 ∈ ω)
6	5	adantl 487	. . . . . 6 ⊢ ((𝜂 ∧ 𝜌) → 𝑖 ∈ ω)
7		bnj570.19	. . . . . . 7 ⊢ (𝜂 ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
8	7, 4	bnj563 35441	. . . . . 6 ⊢ ((𝜂 ∧ 𝜌) → suc 𝑖 ∈ 𝑚)
9	6, 8	jca 521	. . . . 5 ⊢ ((𝜂 ∧ 𝜌) → (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑚))
10		bnj570.30	. . . . . . . 8 ⊢ (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑚 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
11	10	bnj946 35472	. . . . . . 7 ⊢ (𝜓′ ↔ ∀𝑖(𝑖 ∈ ω → (suc 𝑖 ∈ 𝑚 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))))
12		sp 2223	. . . . . . 7 ⊢ (∀𝑖(𝑖 ∈ ω → (suc 𝑖 ∈ 𝑚 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))) → (𝑖 ∈ ω → (suc 𝑖 ∈ 𝑚 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))))
13	11, 12	sylbi 220	. . . . . 6 ⊢ (𝜓′ → (𝑖 ∈ ω → (suc 𝑖 ∈ 𝑚 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))))
14	13	imp32 424	. . . . 5 ⊢ ((𝜓′ ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑚)) → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))
15	3, 9, 14	syl2an 608	. . . 4 ⊢ ((𝜏 ∧ (𝜂 ∧ 𝜌)) → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))
16	1, 15	simplbiim 514	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜌) → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))
17		bnj570.40	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) → 𝐺 Fn 𝑛)
18	17	fnfund 6659	. . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) → Fun 𝐺)
19	18	bnj721 35455	. . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜌) → Fun 𝐺)
20		bnj570.26	. . . . . 6 ⊢ 𝐺 = (𝑓 ∪ {⟨𝑚, 𝐶⟩})
21	20	bnj931 35468	. . . . 5 ⊢ 𝑓 ⊆ 𝐺
22	21	a1i 11	. . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜌) → 𝑓 ⊆ 𝐺)
23		bnj667 35450	. . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜌) → (𝜏 ∧ 𝜂 ∧ 𝜌))
24	2	bnj564 35442	. . . . . . 7 ⊢ (𝜏 → dom 𝑓 = 𝑚)
25		eleq2 2853	. . . . . . . 8 ⊢ (dom 𝑓 = 𝑚 → (suc 𝑖 ∈ dom 𝑓 ↔ suc 𝑖 ∈ 𝑚))
26	25	biimpar 483	. . . . . . 7 ⊢ ((dom 𝑓 = 𝑚 ∧ suc 𝑖 ∈ 𝑚) → suc 𝑖 ∈ dom 𝑓)
27	24, 8, 26	syl2an 608	. . . . . 6 ⊢ ((𝜏 ∧ (𝜂 ∧ 𝜌)) → suc 𝑖 ∈ dom 𝑓)
28	27	3impb 1132	. . . . 5 ⊢ ((𝜏 ∧ 𝜂 ∧ 𝜌) → suc 𝑖 ∈ dom 𝑓)
29	23, 28	syl 18	. . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜌) → suc 𝑖 ∈ dom 𝑓)
30	19, 22, 29	bnj1502 35545	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜌) → (𝐺‘suc 𝑖) = (𝑓‘suc 𝑖))
31	2	simp1bi 1163	. . . . . . . . 9 ⊢ (𝜏 → 𝑓 Fn 𝑚)
32		bnj252 35401	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ↔ (𝑚 ∈ 𝐷 ∧ (𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝)))
33	32	simplbi 502	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) → 𝑚 ∈ 𝐷)
34	7, 33	sylbi 220	. . . . . . . . . . . 12 ⊢ (𝜂 → 𝑚 ∈ 𝐷)
35		eldifi 4070	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ (ω ∖ {∅}) → 𝑚 ∈ ω)
36		bnj570.3	. . . . . . . . . . . . 13 ⊢ 𝐷 = (ω ∖ {∅})
37	35, 36	eleq2s 2882	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ 𝐷 → 𝑚 ∈ ω)
38		nnord 7906	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ω → Ord 𝑚)
39	34, 37, 38	3syl 19	. . . . . . . . . . 11 ⊢ (𝜂 → Ord 𝑚)
40	39	adantr 486	. . . . . . . . . 10 ⊢ ((𝜂 ∧ 𝜌) → Ord 𝑚)
41	40, 8	jca 521	. . . . . . . . 9 ⊢ ((𝜂 ∧ 𝜌) → (Ord 𝑚 ∧ suc 𝑖 ∈ 𝑚))
42	31, 41	anim12i 625	. . . . . . . 8 ⊢ ((𝜏 ∧ (𝜂 ∧ 𝜌)) → (𝑓 Fn 𝑚 ∧ (Ord 𝑚 ∧ suc 𝑖 ∈ 𝑚)))
43		fndm 6661	. . . . . . . . 9 ⊢ (𝑓 Fn 𝑚 → dom 𝑓 = 𝑚)
44		elelsuc 6453	. . . . . . . . . 10 ⊢ (suc 𝑖 ∈ 𝑚 → suc 𝑖 ∈ suc 𝑚)
45		ordsucelsuc 7854	. . . . . . . . . . 11 ⊢ (Ord 𝑚 → (𝑖 ∈ 𝑚 ↔ suc 𝑖 ∈ suc 𝑚))
46	45	biimpar 483	. . . . . . . . . 10 ⊢ ((Ord 𝑚 ∧ suc 𝑖 ∈ suc 𝑚) → 𝑖 ∈ 𝑚)
47	44, 46	sylan2 605	. . . . . . . . 9 ⊢ ((Ord 𝑚 ∧ suc 𝑖 ∈ 𝑚) → 𝑖 ∈ 𝑚)
48	43, 47	anim12i 625	. . . . . . . 8 ⊢ ((𝑓 Fn 𝑚 ∧ (Ord 𝑚 ∧ suc 𝑖 ∈ 𝑚)) → (dom 𝑓 = 𝑚 ∧ 𝑖 ∈ 𝑚))
49		eleq2 2853	. . . . . . . . 9 ⊢ (dom 𝑓 = 𝑚 → (𝑖 ∈ dom 𝑓 ↔ 𝑖 ∈ 𝑚))
50	49	biimpar 483	. . . . . . . 8 ⊢ ((dom 𝑓 = 𝑚 ∧ 𝑖 ∈ 𝑚) → 𝑖 ∈ dom 𝑓)
51	42, 48, 50	3syl 19	. . . . . . 7 ⊢ ((𝜏 ∧ (𝜂 ∧ 𝜌)) → 𝑖 ∈ dom 𝑓)
52	51	3impb 1132	. . . . . 6 ⊢ ((𝜏 ∧ 𝜂 ∧ 𝜌) → 𝑖 ∈ dom 𝑓)
53	23, 52	syl 18	. . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜌) → 𝑖 ∈ dom 𝑓)
54	19, 22, 53	bnj1502 35545	. . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜌) → (𝐺‘𝑖) = (𝑓‘𝑖))
55	54	iuneq1d 4982	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜌) → ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))
56	16, 30, 55	3eqtr4d 2808	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜌) → (𝐺‘suc 𝑖) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅))
57		bnj570.24	. 2 ⊢ 𝐾 = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)
58	56, 57	eqtr4di 2816	1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜌) → (𝐺‘suc 𝑖) = 𝐾)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103  ∀wal 1568   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∀wral 3080   ⊆ wss 3900   ∖ cdif 4048   ∪ cun 4152  ∅c0 4281  {csn 4588  ⟨cop 4838  ∪ ciun 4954  dom cdm 5655  Ord word 6376  suc csuc 6379  Fun wfun 6549   Fn wfn 6550  ‘cfv 6555  ωcom 7898   ∧ w-bnj17 35384   predc-bnj14 35386   FrSe w-bnj15 35390

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-pr 5395  ax-un 7772

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-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-iun 4956  df-br 5107  df-opab 5171  df-tr 5216  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-res 5667  df-ord 6380  df-on 6381  df-suc 6383  df-iota 6509  df-fun 6557  df-fn 6558  df-fv 6563  df-om 7899  df-bnj17 35385

This theorem is used by:  bnj571  35603