Theorem bnj570 34445

Description: Technical lemma for bnj852 34461. 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 34242	. . . 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 1144	. . . . 5 ⊢ (𝜏 → 𝜓′)
4		bnj570.21	. . . . . . . 8 ⊢ (𝜌 ↔ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛 ∧ 𝑚 ≠ suc 𝑖))
5	4	simp1bi 1142	. . . . . . 7 ⊢ (𝜌 → 𝑖 ∈ ω)
6	5	adantl 481	. . . . . 6 ⊢ ((𝜂 ∧ 𝜌) → 𝑖 ∈ ω)
7		bnj570.19	. . . . . . 7 ⊢ (𝜂 ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
8	7, 4	bnj563 34283	. . . . . 6 ⊢ ((𝜂 ∧ 𝜌) → suc 𝑖 ∈ 𝑚)
9	6, 8	jca 511	. . . . 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 34314	. . . . . . 7 ⊢ (𝜓′ ↔ ∀𝑖(𝑖 ∈ ω → (suc 𝑖 ∈ 𝑚 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))))
12		sp 2168	. . . . . . 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 216	. . . . . 6 ⊢ (𝜓′ → (𝑖 ∈ ω → (suc 𝑖 ∈ 𝑚 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))))
14	13	imp32 418	. . . . 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 504	. . 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 6643	. . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) → Fun 𝐺)
19	18	bnj721 34297	. . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜌) → Fun 𝐺)
20		bnj570.26	. . . . . 6 ⊢ 𝐺 = (𝑓 ∪ {⟨𝑚, 𝐶⟩})
21	20	bnj931 34310	. . . . 5 ⊢ 𝑓 ⊆ 𝐺
22	21	a1i 11	. . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜌) → 𝑓 ⊆ 𝐺)
23		bnj667 34292	. . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜌) → (𝜏 ∧ 𝜂 ∧ 𝜌))
24	2	bnj564 34284	. . . . . . 7 ⊢ (𝜏 → dom 𝑓 = 𝑚)
25		eleq2 2816	. . . . . . . 8 ⊢ (dom 𝑓 = 𝑚 → (suc 𝑖 ∈ dom 𝑓 ↔ suc 𝑖 ∈ 𝑚))
26	25	biimpar 477	. . . . . . 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 1112	. . . . 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 34388	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜌) → (𝐺‘suc 𝑖) = (𝑓‘suc 𝑖))
31	2	simp1bi 1142	. . . . . . . . 9 ⊢ (𝜏 → 𝑓 Fn 𝑚)
32		bnj252 34243	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ↔ (𝑚 ∈ 𝐷 ∧ (𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝)))
33	32	simplbi 497	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) → 𝑚 ∈ 𝐷)
34	7, 33	sylbi 216	. . . . . . . . . . . 12 ⊢ (𝜂 → 𝑚 ∈ 𝐷)
35		eldifi 4121	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ (ω ∖ {∅}) → 𝑚 ∈ ω)
36		bnj570.3	. . . . . . . . . . . . 13 ⊢ 𝐷 = (ω ∖ {∅})
37	35, 36	eleq2s 2845	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ 𝐷 → 𝑚 ∈ ω)
38		nnord 7859	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ω → Ord 𝑚)
39	34, 37, 38	3syl 18	. . . . . . . . . . 11 ⊢ (𝜂 → Ord 𝑚)
40	39	adantr 480	. . . . . . . . . 10 ⊢ ((𝜂 ∧ 𝜌) → Ord 𝑚)
41	40, 8	jca 511	. . . . . . . . 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 6645	. . . . . . . . 9 ⊢ (𝑓 Fn 𝑚 → dom 𝑓 = 𝑚)
44		elelsuc 6430	. . . . . . . . . 10 ⊢ (suc 𝑖 ∈ 𝑚 → suc 𝑖 ∈ suc 𝑚)
45		ordsucelsuc 7806	. . . . . . . . . . 11 ⊢ (Ord 𝑚 → (𝑖 ∈ 𝑚 ↔ suc 𝑖 ∈ suc 𝑚))
46	45	biimpar 477	. . . . . . . . . 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 2816	. . . . . . . . 9 ⊢ (dom 𝑓 = 𝑚 → (𝑖 ∈ dom 𝑓 ↔ 𝑖 ∈ 𝑚))
50	49	biimpar 477	. . . . . . . 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 1112	. . . . . 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 34388	. . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜌) → (𝐺‘𝑖) = (𝑓‘𝑖))
55	54	iuneq1d 5017	. . 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 2776	. 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 2784	1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜌) → (𝐺‘suc 𝑖) = 𝐾)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1084  ∀wal 1531   = wceq 1533   ∈ wcel 2098   ≠ wne 2934  ∀wral 3055   ∖ cdif 3940   ∪ cun 3941   ⊆ wss 3943  ∅c0 4317  {csn 4623  ⟨cop 4629  ∪ ciun 4990  dom cdm 5669  Ord word 6356  suc csuc 6359  Fun wfun 6530   Fn wfn 6531  ‘cfv 6536  ωcom 7851   ∧ w-bnj17 34226   predc-bnj14 34228   FrSe w-bnj15 34232

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-res 5681  df-ord 6360  df-on 6361  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-fv 6544  df-om 7852  df-bnj17 34227

This theorem is referenced by:  bnj571  34446