Theorem bnj570 32552

Description: Technical lemma for bnj852 32568. 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 32347	. . . 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 1149	. . . . 5 ⊢ (𝜏 → 𝜓′)
4		bnj570.21	. . . . . . . 8 ⊢ (𝜌 ↔ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛 ∧ 𝑚 ≠ suc 𝑖))
5	4	simp1bi 1147	. . . . . . 7 ⊢ (𝜌 → 𝑖 ∈ ω)
6	5	adantl 485	. . . . . 6 ⊢ ((𝜂 ∧ 𝜌) → 𝑖 ∈ ω)
7		bnj570.19	. . . . . . 7 ⊢ (𝜂 ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
8	7, 4	bnj563 32389	. . . . . 6 ⊢ ((𝜂 ∧ 𝜌) → suc 𝑖 ∈ 𝑚)
9	6, 8	jca 515	. . . . 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 32421	. . . . . . 7 ⊢ (𝜓′ ↔ ∀𝑖(𝑖 ∈ ω → (suc 𝑖 ∈ 𝑚 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))))
12		sp 2182	. . . . . . 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 422	. . . . 5 ⊢ ((𝜓′ ∧ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑚)) → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))
15	3, 9, 14	syl2an 599	. . . 4 ⊢ ((𝜏 ∧ (𝜂 ∧ 𝜌)) → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))
16	1, 15	simplbiim 508	. . 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 32416	. . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) → Fun 𝐺)
19	18	bnj721 32403	. . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜌) → Fun 𝐺)
20		bnj570.26	. . . . . 6 ⊢ 𝐺 = (𝑓 ∪ {⟨𝑚, 𝐶⟩})
21	20	bnj931 32417	. . . . 5 ⊢ 𝑓 ⊆ 𝐺
22	21	a1i 11	. . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜌) → 𝑓 ⊆ 𝐺)
23		bnj667 32398	. . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜌) → (𝜏 ∧ 𝜂 ∧ 𝜌))
24	2	bnj564 32390	. . . . . . 7 ⊢ (𝜏 → dom 𝑓 = 𝑚)
25		eleq2 2819	. . . . . . . 8 ⊢ (dom 𝑓 = 𝑚 → (suc 𝑖 ∈ dom 𝑓 ↔ suc 𝑖 ∈ 𝑚))
26	25	biimpar 481	. . . . . . 7 ⊢ ((dom 𝑓 = 𝑚 ∧ suc 𝑖 ∈ 𝑚) → suc 𝑖 ∈ dom 𝑓)
27	24, 8, 26	syl2an 599	. . . . . 6 ⊢ ((𝜏 ∧ (𝜂 ∧ 𝜌)) → suc 𝑖 ∈ dom 𝑓)
28	27	3impb 1117	. . . . 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 32495	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜌) → (𝐺‘suc 𝑖) = (𝑓‘suc 𝑖))
31	2	simp1bi 1147	. . . . . . . . 9 ⊢ (𝜏 → 𝑓 Fn 𝑚)
32		bnj252 32348	. . . . . . . . . . . . . 14 ⊢ ((𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) ↔ (𝑚 ∈ 𝐷 ∧ (𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝)))
33	32	simplbi 501	. . . . . . . . . . . . 13 ⊢ ((𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝) → 𝑚 ∈ 𝐷)
34	7, 33	sylbi 220	. . . . . . . . . . . 12 ⊢ (𝜂 → 𝑚 ∈ 𝐷)
35		eldifi 4027	. . . . . . . . . . . . 13 ⊢ (𝑚 ∈ (ω ∖ {∅}) → 𝑚 ∈ ω)
36		bnj570.3	. . . . . . . . . . . . 13 ⊢ 𝐷 = (ω ∖ {∅})
37	35, 36	eleq2s 2849	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ 𝐷 → 𝑚 ∈ ω)
38		nnord 7630	. . . . . . . . . . . 12 ⊢ (𝑚 ∈ ω → Ord 𝑚)
39	34, 37, 38	3syl 18	. . . . . . . . . . 11 ⊢ (𝜂 → Ord 𝑚)
40	39	adantr 484	. . . . . . . . . 10 ⊢ ((𝜂 ∧ 𝜌) → Ord 𝑚)
41	40, 8	jca 515	. . . . . . . . 9 ⊢ ((𝜂 ∧ 𝜌) → (Ord 𝑚 ∧ suc 𝑖 ∈ 𝑚))
42	31, 41	anim12i 616	. . . . . . . 8 ⊢ ((𝜏 ∧ (𝜂 ∧ 𝜌)) → (𝑓 Fn 𝑚 ∧ (Ord 𝑚 ∧ suc 𝑖 ∈ 𝑚)))
43		fndm 6459	. . . . . . . . 9 ⊢ (𝑓 Fn 𝑚 → dom 𝑓 = 𝑚)
44		elelsuc 6263	. . . . . . . . . 10 ⊢ (suc 𝑖 ∈ 𝑚 → suc 𝑖 ∈ suc 𝑚)
45		ordsucelsuc 7579	. . . . . . . . . . 11 ⊢ (Ord 𝑚 → (𝑖 ∈ 𝑚 ↔ suc 𝑖 ∈ suc 𝑚))
46	45	biimpar 481	. . . . . . . . . 10 ⊢ ((Ord 𝑚 ∧ suc 𝑖 ∈ suc 𝑚) → 𝑖 ∈ 𝑚)
47	44, 46	sylan2 596	. . . . . . . . 9 ⊢ ((Ord 𝑚 ∧ suc 𝑖 ∈ 𝑚) → 𝑖 ∈ 𝑚)
48	43, 47	anim12i 616	. . . . . . . 8 ⊢ ((𝑓 Fn 𝑚 ∧ (Ord 𝑚 ∧ suc 𝑖 ∈ 𝑚)) → (dom 𝑓 = 𝑚 ∧ 𝑖 ∈ 𝑚))
49		eleq2 2819	. . . . . . . . 9 ⊢ (dom 𝑓 = 𝑚 → (𝑖 ∈ dom 𝑓 ↔ 𝑖 ∈ 𝑚))
50	49	biimpar 481	. . . . . . . 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 1117	. . . . . 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 32495	. . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜌) → (𝐺‘𝑖) = (𝑓‘𝑖))
55	54	iuneq1d 4917	. . 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 2781	. 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 2789	1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜌) → (𝐺‘suc 𝑖) = 𝐾)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1089  ∀wal 1541   = wceq 1543   ∈ wcel 2112   ≠ wne 2932  ∀wral 3051   ∖ cdif 3850   ∪ cun 3851   ⊆ wss 3853  ∅c0 4223  {csn 4527  ⟨cop 4533  ∪ ciun 4890  dom cdm 5536  Ord word 6190  suc csuc 6193  Fun wfun 6352   Fn wfn 6353  ‘cfv 6358  ωcom 7622   ∧ w-bnj17 32331   predc-bnj14 32333   FrSe w-bnj15 32337

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 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pr 5307  ax-un 7501

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3400  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4224  df-if 4426  df-sn 4528  df-pr 4530  df-tp 4532  df-op 4534  df-uni 4806  df-iun 4892  df-br 5040  df-opab 5102  df-tr 5147  df-id 5440  df-eprel 5445  df-po 5453  df-so 5454  df-fr 5494  df-we 5496  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-res 5548  df-ord 6194  df-on 6195  df-suc 6197  df-iota 6316  df-fun 6360  df-fn 6361  df-fv 6366  df-om 7623  df-bnj17 32332

This theorem is referenced by:  bnj571  32553