Theorem bnj570 31789

Description: Technical lemma for bnj852 31805. 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 31585	. . . 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 1140	. . . . 5 ⊢ (𝜏 → 𝜓′)
4		bnj570.21	. . . . . . . 8 ⊢ (𝜌 ↔ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛 ∧ 𝑚 ≠ suc 𝑖))
5	4	simp1bi 1138	. . . . . . 7 ⊢ (𝜌 → 𝑖 ∈ ω)
6	5	adantl 482	. . . . . 6 ⊢ ((𝜂 ∧ 𝜌) → 𝑖 ∈ ω)
7		bnj570.19	. . . . . . 7 ⊢ (𝜂 ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
8	7, 4	bnj563 31627	. . . . . 6 ⊢ ((𝜂 ∧ 𝜌) → suc 𝑖 ∈ 𝑚)
9	6, 8	jca 512	. . . . 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 31659	. . . . . . 7 ⊢ (𝜓′ ↔ ∀𝑖(𝑖 ∈ ω → (suc 𝑖 ∈ 𝑚 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))))
12		sp 2148	. . . . . . 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 218	. . . . . 6 ⊢ (𝜓′ → (𝑖 ∈ ω → (suc 𝑖 ∈ 𝑚 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))))
14	13	imp32 419	. . . . 5 ⊢ ((𝜓′ ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑚)) → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))
15	3, 9, 14	syl2an 595	. . . 4 ⊢ ((𝜏 ∧ (𝜂 ∧ 𝜌)) → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))
16	1, 15	simplbiim 505	. . 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	bnj930 31654	. . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) → Fun 𝐺)
19	18	bnj721 31641	. . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜌) → Fun 𝐺)
20		bnj570.26	. . . . . 6 ⊢ 𝐺 = (𝑓 ∪ {⟨𝑚, 𝐶⟩})
21	20	bnj931 31655	. . . . 5 ⊢ 𝑓 ⊆ 𝐺
22	21	a1i 11	. . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜌) → 𝑓 ⊆ 𝐺)
23		bnj667 31636	. . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜌) → (𝜏 ∧ 𝜂 ∧ 𝜌))
24	2	bnj564 31628	. . . . . . 7 ⊢ (𝜏 → dom 𝑓 = 𝑚)
25		eleq2 2873	. . . . . . . 8 ⊢ (dom 𝑓 = 𝑚 → (suc 𝑖 ∈ dom 𝑓 ↔ suc 𝑖 ∈ 𝑚))
26	25	biimpar 478	. . . . . . 7 ⊢ ((dom 𝑓 = 𝑚 ∧ suc 𝑖 ∈ 𝑚) → suc 𝑖 ∈ dom 𝑓)
27	24, 8, 26	syl2an 595	. . . . . 6 ⊢ ((𝜏 ∧ (𝜂 ∧ 𝜌)) → suc 𝑖 ∈ dom 𝑓)
28	27	3impb 1108	. . . . 5 ⊢ ((𝜏 ∧ 𝜂 ∧ 𝜌) → suc 𝑖 ∈ dom 𝑓)
29	23, 28	syl 17	. . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜌) → suc 𝑖 ∈ dom 𝑓)
30	19, 22, 29	bnj1502 31732	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜌) → (𝐺‘suc 𝑖) = (𝑓‘suc 𝑖))
31	2	simp1bi 1138	. . . . . . . . 9 ⊢ (𝜏 → 𝑓 Fn 𝑚)
32		bnj252 31586	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ↔ (𝑚 ∈ 𝐷 ∧ (𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝)))
33	32	simplbi 498	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) → 𝑚 ∈ 𝐷)
34	7, 33	sylbi 218	. . . . . . . . . . . 12 ⊢ (𝜂 → 𝑚 ∈ 𝐷)
35		eldifi 4030	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ (ω ∖ {∅}) → 𝑚 ∈ ω)
36		bnj570.3	. . . . . . . . . . . . 13 ⊢ 𝐷 = (ω ∖ {∅})
37	35, 36	eleq2s 2903	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ 𝐷 → 𝑚 ∈ ω)
38		nnord 7451	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ω → Ord 𝑚)
39	34, 37, 38	3syl 18	. . . . . . . . . . 11 ⊢ (𝜂 → Ord 𝑚)
40	39	adantr 481	. . . . . . . . . 10 ⊢ ((𝜂 ∧ 𝜌) → Ord 𝑚)
41	40, 8	jca 512	. . . . . . . . 9 ⊢ ((𝜂 ∧ 𝜌) → (Ord 𝑚 ∧ suc 𝑖 ∈ 𝑚))
42	31, 41	anim12i 612	. . . . . . . 8 ⊢ ((𝜏 ∧ (𝜂 ∧ 𝜌)) → (𝑓 Fn 𝑚 ∧ (Ord 𝑚 ∧ suc 𝑖 ∈ 𝑚)))
43		fndm 6332	. . . . . . . . 9 ⊢ (𝑓 Fn 𝑚 → dom 𝑓 = 𝑚)
44		elelsuc 6145	. . . . . . . . . 10 ⊢ (suc 𝑖 ∈ 𝑚 → suc 𝑖 ∈ suc 𝑚)
45		ordsucelsuc 7400	. . . . . . . . . . 11 ⊢ (Ord 𝑚 → (𝑖 ∈ 𝑚 ↔ suc 𝑖 ∈ suc 𝑚))
46	45	biimpar 478	. . . . . . . . . 10 ⊢ ((Ord 𝑚 ∧ suc 𝑖 ∈ suc 𝑚) → 𝑖 ∈ 𝑚)
47	44, 46	sylan2 592	. . . . . . . . 9 ⊢ ((Ord 𝑚 ∧ suc 𝑖 ∈ 𝑚) → 𝑖 ∈ 𝑚)
48	43, 47	anim12i 612	. . . . . . . 8 ⊢ ((𝑓 Fn 𝑚 ∧ (Ord 𝑚 ∧ suc 𝑖 ∈ 𝑚)) → (dom 𝑓 = 𝑚 ∧ 𝑖 ∈ 𝑚))
49		eleq2 2873	. . . . . . . . 9 ⊢ (dom 𝑓 = 𝑚 → (𝑖 ∈ dom 𝑓 ↔ 𝑖 ∈ 𝑚))
50	49	biimpar 478	. . . . . . . 8 ⊢ ((dom 𝑓 = 𝑚 ∧ 𝑖 ∈ 𝑚) → 𝑖 ∈ dom 𝑓)
51	42, 48, 50	3syl 18	. . . . . . 7 ⊢ ((𝜏 ∧ (𝜂 ∧ 𝜌)) → 𝑖 ∈ dom 𝑓)
52	51	3impb 1108	. . . . . 6 ⊢ ((𝜏 ∧ 𝜂 ∧ 𝜌) → 𝑖 ∈ dom 𝑓)
53	23, 52	syl 17	. . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜌) → 𝑖 ∈ dom 𝑓)
54	19, 22, 53	bnj1502 31732	. . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜌) → (𝐺‘𝑖) = (𝑓‘𝑖))
55	54	iuneq1d 4857	. . 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 2843	. 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	syl6eqr 2851	1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜌) → (𝐺‘suc 𝑖) = 𝐾)

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1080  ∀wal 1523   = wceq 1525   ∈ wcel 2083   ≠ wne 2986  ∀wral 3107   ∖ cdif 3862   ∪ cun 3863   ⊆ wss 3865  ∅c0 4217  {csn 4478  ⟨cop 4484  ∪ ciun 4831  dom cdm 5450  Ord word 6072  suc csuc 6075  Fun wfun 6226   Fn wfn 6227  ‘cfv 6232  ωcom 7443   ∧ w-bnj17 31569   predc-bnj14 31571   FrSe w-bnj15 31575

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-sep 5101  ax-nul 5108  ax-pr 5228  ax-un 7326

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3442  df-sbc 3712  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-pss 3882  df-nul 4218  df-if 4388  df-sn 4479  df-pr 4481  df-tp 4483  df-op 4485  df-uni 4752  df-iun 4833  df-br 4969  df-opab 5031  df-tr 5071  df-id 5355  df-eprel 5360  df-po 5369  df-so 5370  df-fr 5409  df-we 5411  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-res 5462  df-ord 6076  df-on 6077  df-lim 6078  df-suc 6079  df-iota 6196  df-fun 6234  df-fn 6235  df-fv 6240  df-om 7444  df-bnj17 31570

This theorem is referenced by:  bnj571  31790