Theorem bnj557 35234

Description: Technical lemma for bnj852 35254. 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
bnj557.3	⊢ 𝐷 = (ω ∖ {∅})
bnj557.16	⊢ 𝐺 = (𝑓 ∪ {⟨𝑚, ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)⟩})
bnj557.17	⊢ (𝜏 ↔ (𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′))
bnj557.18	⊢ (𝜎 ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚))
bnj557.19	⊢ (𝜂 ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
bnj557.20	⊢ (𝜁 ↔ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛 ∧ 𝑚 = suc 𝑖))
bnj557.21	⊢ 𝐵 = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)
bnj557.22	⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)
bnj557.23	⊢ 𝐾 = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)
bnj557.24	⊢ 𝐿 = ∪ 𝑦 ∈ (𝐺‘𝑝) pred(𝑦, 𝐴, 𝑅)
bnj557.25	⊢ 𝐺 = (𝑓 ∪ {⟨𝑚, 𝐶⟩})
bnj557.28	⊢ (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
bnj557.29	⊢ (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑚 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj557.36	⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → 𝐺 Fn 𝑛)

Assertion

Ref	Expression
bnj557	⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜁) → (𝐺‘𝑚) = 𝐿)

Distinct variable groups:   𝐴,𝑖,𝑝,𝑦   𝑦,𝐺   𝑅,𝑖,𝑝,𝑦   𝑓,𝑖,𝑝,𝑦   𝑖,𝑚,𝑝   𝑝,𝜑′

Allowed substitution hints:   𝜏(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜂(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜁(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜎(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐴(𝑥,𝑓,𝑚,𝑛)   𝐵(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐶(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐷(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝑅(𝑥,𝑓,𝑚,𝑛)   𝐺(𝑥,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐾(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐿(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜑′(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛)   𝜓′(𝑥,𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)

Proof of Theorem bnj557

Step	Hyp	Ref	Expression
1		3an4anass 1120	. . . . 5 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) ∧ 𝜁) ↔ ((𝑅 FrSe 𝐴 ∧ 𝜏) ∧ (𝜂 ∧ 𝜁)))
2		bnj557.18	. . . . . . . 8 ⊢ (𝜎 ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚))
3		bnj557.19	. . . . . . . 8 ⊢ (𝜂 ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ ω ∧ 𝑚 = suc 𝑝))
4	2, 3	bnj556 35233	. . . . . . 7 ⊢ (𝜂 → 𝜎)
5	4	3anim3i 1170	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) → (𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎))
6		bnj557.20	. . . . . . 7 ⊢ (𝜁 ↔ (𝑖 ∈ ω ∧ suc 𝑖 ∈ 𝑛 ∧ 𝑚 = suc 𝑖))
7		vex 3465	. . . . . . . 8 ⊢ 𝑖 ∈ V
8	7	bnj216 35066	. . . . . . 7 ⊢ (𝑚 = suc 𝑖 → 𝑖 ∈ 𝑚)
9	6, 8	bnj837 35095	. . . . . 6 ⊢ (𝜁 → 𝑖 ∈ 𝑚)
10	5, 9	anim12i 624	. . . . 5 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂) ∧ 𝜁) → ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚))
11	1, 10	sylbir 238	. . . 4 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏) ∧ (𝜂 ∧ 𝜁)) → ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚))
12	3	bnj1254 35142	. . . . . 6 ⊢ (𝜂 → 𝑚 = suc 𝑝)
13	6	simp3bi 1163	. . . . . 6 ⊢ (𝜁 → 𝑚 = suc 𝑖)
14		bnj551 35076	. . . . . 6 ⊢ ((𝑚 = suc 𝑝 ∧ 𝑚 = suc 𝑖) → 𝑝 = 𝑖)
15	12, 13, 14	syl2an 607	. . . . 5 ⊢ ((𝜂 ∧ 𝜁) → 𝑝 = 𝑖)
16	15	adantl 486	. . . 4 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏) ∧ (𝜂 ∧ 𝜁)) → 𝑝 = 𝑖)
17	11, 16	jca 520	. . 3 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏) ∧ (𝜂 ∧ 𝜁)) → (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚) ∧ 𝑝 = 𝑖))
18		bnj256 35040	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜁) ↔ ((𝑅 FrSe 𝐴 ∧ 𝜏) ∧ (𝜂 ∧ 𝜁)))
19		df-3an 1103	. . 3 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚 ∧ 𝑝 = 𝑖) ↔ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚) ∧ 𝑝 = 𝑖))
20	17, 18, 19	3imtr4i 295	. 2 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜁) → ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚 ∧ 𝑝 = 𝑖))
21		bnj557.28	. . 3 ⊢ (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
22		bnj557.29	. . 3 ⊢ (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑚 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
23		bnj557.3	. . 3 ⊢ 𝐷 = (ω ∖ {∅})
24		bnj557.16	. . 3 ⊢ 𝐺 = (𝑓 ∪ {⟨𝑚, ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)⟩})
25		bnj557.17	. . 3 ⊢ (𝜏 ↔ (𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′))
26		bnj557.22	. . 3 ⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)
27		bnj557.25	. . 3 ⊢ 𝐺 = (𝑓 ∪ {⟨𝑚, 𝐶⟩})
28		bnj557.21	. . 3 ⊢ 𝐵 = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)
29		bnj557.23	. . 3 ⊢ 𝐾 = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)
30		bnj557.24	. . 3 ⊢ 𝐿 = ∪ 𝑦 ∈ (𝐺‘𝑝) pred(𝑦, 𝐴, 𝑅)
31		bnj557.36	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → 𝐺 Fn 𝑛)
32	21, 22, 23, 24, 25, 2, 26, 27, 28, 29, 30, 31	bnj553 35231	. 2 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚 ∧ 𝑝 = 𝑖) → (𝐺‘𝑚) = 𝐿)
33	20, 32	syl 18	1 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜂 ∧ 𝜁) → (𝐺‘𝑚) = 𝐿)

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1567   ∈ wcel 2149  ∀wral 3085   ∖ cdif 3908   ∪ cun 3909  ∅c0 4292  {csn 4592  ⟨cop 4598  ∪ ciun 4958  suc csuc 6363   Fn wfn 6532  ‘cfv 6537  ωcom 7862   ∧ w-bnj17 35020   predc-bnj14 35022   FrSe w-bnj15 35026

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-12 2219  ax-ext 2741  ax-sep 5259  ax-pr 5405  ax-un 7733  ax-reg 9554

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3423  df-v 3463  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4491  df-pw 4567  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-iun 4960  df-br 5112  df-opab 5176  df-id 5557  df-eprel 5562  df-fr 5615  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-res 5674  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-fv 6545  df-bnj17 35021

This theorem is referenced by:  bnj558  35235