Theorem bnj553 35528

Description: Technical lemma for bnj852 35551. 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
bnj553.1	⊢ (𝜑′ ↔ (𝑓‘∅) = pred(𝑥, 𝐴, 𝑅))
bnj553.2	⊢ (𝜓′ ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑚 → (𝑓‘suc 𝑖) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj553.3	⊢ 𝐷 = (ω ∖ {∅})
bnj553.4	⊢ 𝐺 = (𝑓 ∪ {⟨𝑚, ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)⟩})
bnj553.5	⊢ (𝜏 ↔ (𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′))
bnj553.6	⊢ (𝜎 ↔ (𝑚 ∈ 𝐷 ∧ 𝑛 = suc 𝑚 ∧ 𝑝 ∈ 𝑚))
bnj553.7	⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)
bnj553.8	⊢ 𝐺 = (𝑓 ∪ {⟨𝑚, 𝐶⟩})
bnj553.9	⊢ 𝐵 = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)
bnj553.10	⊢ 𝐾 = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)
bnj553.11	⊢ 𝐿 = ∪ 𝑦 ∈ (𝐺‘𝑝) pred(𝑦, 𝐴, 𝑅)
bnj553.12	⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → 𝐺 Fn 𝑛)

Assertion

Ref	Expression
bnj553	⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚 ∧ 𝑝 = 𝑖) → (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)   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 bnj553

Step	Hyp	Ref	Expression
1		bnj553.12	. . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → 𝐺 Fn 𝑛)
2	1	fnfund 6640	. . . 4 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → Fun 𝐺)
3		opex 5431	. . . . . . 7 ⊢ ⟨𝑚, 𝐶⟩ ∈ V
4	3	snid 4622	. . . . . 6 ⊢ ⟨𝑚, 𝐶⟩ ∈ {⟨𝑚, 𝐶⟩}
5		elun2 4181	. . . . . 6 ⊢ (⟨𝑚, 𝐶⟩ ∈ {⟨𝑚, 𝐶⟩} → ⟨𝑚, 𝐶⟩ ∈ (𝑓 ∪ {⟨𝑚, 𝐶⟩}))
6	4, 5	ax-mp 5	. . . . 5 ⊢ ⟨𝑚, 𝐶⟩ ∈ (𝑓 ∪ {⟨𝑚, 𝐶⟩})
7		bnj553.8	. . . . 5 ⊢ 𝐺 = (𝑓 ∪ {⟨𝑚, 𝐶⟩})
8	6, 7	eleqtrri 2859	. . . 4 ⊢ ⟨𝑚, 𝐶⟩ ∈ 𝐺
9		funopfv 6934	. . . 4 ⊢ (Fun 𝐺 → (⟨𝑚, 𝐶⟩ ∈ 𝐺 → (𝐺‘𝑚) = 𝐶))
10	2, 8, 9	mpisyl 22	. . 3 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) → (𝐺‘𝑚) = 𝐶)
11	10	3ad2ant1 1151	. 2 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚 ∧ 𝑝 = 𝑖) → (𝐺‘𝑚) = 𝐶)
12		fveq2 6885	. . . . . 6 ⊢ (𝑝 = 𝑖 → (𝐺‘𝑝) = (𝐺‘𝑖))
13	12	bnj1113 35416	. . . . 5 ⊢ (𝑝 = 𝑖 → ∪ 𝑦 ∈ (𝐺‘𝑝) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅))
14		bnj553.11	. . . . 5 ⊢ 𝐿 = ∪ 𝑦 ∈ (𝐺‘𝑝) pred(𝑦, 𝐴, 𝑅)
15		bnj553.10	. . . . 5 ⊢ 𝐾 = ∪ 𝑦 ∈ (𝐺‘𝑖) pred(𝑦, 𝐴, 𝑅)
16	13, 14, 15	3eqtr4g 2820	. . . 4 ⊢ (𝑝 = 𝑖 → 𝐿 = 𝐾)
17	16	3ad2ant3 1153	. . 3 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚 ∧ 𝑝 = 𝑖) → 𝐿 = 𝐾)
18		bnj553.5	. . . . 5 ⊢ (𝜏 ↔ (𝑓 Fn 𝑚 ∧ 𝜑′ ∧ 𝜓′))
19		bnj553.9	. . . . 5 ⊢ 𝐵 = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅)
20		bnj553.4	. . . . 5 ⊢ 𝐺 = (𝑓 ∪ {⟨𝑚, ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)⟩})
21	18, 19, 15, 20, 1	bnj548 35527	. . . 4 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚) → 𝐵 = 𝐾)
22	21	3adant3 1150	. . 3 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚 ∧ 𝑝 = 𝑖) → 𝐵 = 𝐾)
23		fveq2 6885	. . . . . 6 ⊢ (𝑝 = 𝑖 → (𝑓‘𝑝) = (𝑓‘𝑖))
24	23	bnj1113 35416	. . . . 5 ⊢ (𝑝 = 𝑖 → ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))
25		bnj553.7	. . . . . . 7 ⊢ 𝐶 = ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅)
26	19, 25	eqeq12i 2778	. . . . . 6 ⊢ (𝐵 = 𝐶 ↔ ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅))
27		eqcom 2767	. . . . . 6 ⊢ (∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) ↔ ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))
28	26, 27	bitri 278	. . . . 5 ⊢ (𝐵 = 𝐶 ↔ ∪ 𝑦 ∈ (𝑓‘𝑝) pred(𝑦, 𝐴, 𝑅) = ∪ 𝑦 ∈ (𝑓‘𝑖) pred(𝑦, 𝐴, 𝑅))
29	24, 28	sylibr 237	. . . 4 ⊢ (𝑝 = 𝑖 → 𝐵 = 𝐶)
30	29	3ad2ant3 1153	. . 3 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚 ∧ 𝑝 = 𝑖) → 𝐵 = 𝐶)
31	17, 22, 30	3eqtr2rd 2802	. 2 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚 ∧ 𝑝 = 𝑖) → 𝐶 = 𝐿)
32	11, 31	eqtrd 2795	1 ⊢ (((𝑅 FrSe 𝐴 ∧ 𝜏 ∧ 𝜎) ∧ 𝑖 ∈ 𝑚 ∧ 𝑝 = 𝑖) → (𝐺‘𝑚) = 𝐿)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∀wral 3076   ∖ cdif 4045   ∪ cun 4149  ∅c0 4278  {csn 4583  ⟨cop 4589  ∪ ciun 4950  suc csuc 6364  Fun wfun 6532   Fn wfn 6533  ‘cfv 6538  ωcom 7877   predc-bnj14 35319   FrSe w-bnj15 35323

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-res 5662  df-iota 6494  df-fun 6540  df-fn 6541  df-fv 6546

This theorem is used by:  bnj557  35531