Theorem bnj1449 34358

Description: Technical lemma for bnj60 34372. 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
bnj1449.1	⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1449.2	⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1449.3	⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
bnj1449.4	⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
bnj1449.5	⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}
bnj1449.6	⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))
bnj1449.7	⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))
bnj1449.8	⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)
bnj1449.9	⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
bnj1449.10	⊢ 𝑃 = ∪ 𝐻
bnj1449.11	⊢ 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1449.12	⊢ 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})
bnj1449.13	⊢ 𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩
bnj1449.14	⊢ 𝐸 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
bnj1449.15	⊢ (𝜒 → 𝑃 Fn trCl(𝑥, 𝐴, 𝑅))
bnj1449.16	⊢ (𝜒 → 𝑄 Fn ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
bnj1449.17	⊢ (𝜃 ↔ (𝜒 ∧ 𝑧 ∈ 𝐸))
bnj1449.18	⊢ (𝜂 ↔ (𝜃 ∧ 𝑧 ∈ {𝑥}))
bnj1449.19	⊢ (𝜁 ↔ (𝜃 ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)))

Assertion

Ref	Expression
bnj1449	⊢ (𝜁 → ∀𝑓𝜁)

Distinct variable groups:   𝐴,𝑓   𝑓,𝐸   𝑅,𝑓   𝑥,𝑓   𝑦,𝑓   𝑧,𝑓

Allowed substitution hints:   𝜓(𝑥,𝑦,𝑧,𝑓,𝑑)   𝜒(𝑥,𝑦,𝑧,𝑓,𝑑)   𝜃(𝑥,𝑦,𝑧,𝑓,𝑑)   𝜏(𝑥,𝑦,𝑧,𝑓,𝑑)   𝜂(𝑥,𝑦,𝑧,𝑓,𝑑)   𝜁(𝑥,𝑦,𝑧,𝑓,𝑑)   𝐴(𝑥,𝑦,𝑧,𝑑)   𝐵(𝑥,𝑦,𝑧,𝑓,𝑑)   𝐶(𝑥,𝑦,𝑧,𝑓,𝑑)   𝐷(𝑥,𝑦,𝑧,𝑓,𝑑)   𝑃(𝑥,𝑦,𝑧,𝑓,𝑑)   𝑄(𝑥,𝑦,𝑧,𝑓,𝑑)   𝑅(𝑥,𝑦,𝑧,𝑑)   𝐸(𝑥,𝑦,𝑧,𝑑)   𝐺(𝑥,𝑦,𝑧,𝑓,𝑑)   𝐻(𝑥,𝑦,𝑧,𝑓,𝑑)   𝑊(𝑥,𝑦,𝑧,𝑓,𝑑)   𝑌(𝑥,𝑦,𝑧,𝑓,𝑑)   𝑍(𝑥,𝑦,𝑧,𝑓,𝑑)   𝜏′(𝑥,𝑦,𝑧,𝑓,𝑑)

Proof of Theorem bnj1449

Step	Hyp	Ref	Expression
1		bnj1449.19	. . 3 ⊢ (𝜁 ↔ (𝜃 ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)))
2		bnj1449.17	. . . . 5 ⊢ (𝜃 ↔ (𝜒 ∧ 𝑧 ∈ 𝐸))
3		bnj1449.7	. . . . . . 7 ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))
4		bnj1449.6	. . . . . . . . 9 ⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))
5		nfv 1916	. . . . . . . . . 10 ⊢ Ⅎ𝑓 𝑅 FrSe 𝐴
6		bnj1449.5	. . . . . . . . . . . 12 ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}
7		nfe1 2146	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑓∃𝑓𝜏
8	7	nfn 1859	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑓 ¬ ∃𝑓𝜏
9		nfcv 2902	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑓𝐴
10	8, 9	nfrabw 3467	. . . . . . . . . . . 12 ⊢ Ⅎ𝑓{𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}
11	6, 10	nfcxfr 2900	. . . . . . . . . . 11 ⊢ Ⅎ𝑓𝐷
12		nfcv 2902	. . . . . . . . . . 11 ⊢ Ⅎ𝑓∅
13	11, 12	nfne 3042	. . . . . . . . . 10 ⊢ Ⅎ𝑓 𝐷 ≠ ∅
14	5, 13	nfan 1901	. . . . . . . . 9 ⊢ Ⅎ𝑓(𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅)
15	4, 14	nfxfr 1854	. . . . . . . 8 ⊢ Ⅎ𝑓𝜓
16	11	nfcri 2889	. . . . . . . 8 ⊢ Ⅎ𝑓 𝑥 ∈ 𝐷
17		nfv 1916	. . . . . . . . 9 ⊢ Ⅎ𝑓 ¬ 𝑦𝑅𝑥
18	11, 17	nfralw 3307	. . . . . . . 8 ⊢ Ⅎ𝑓∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥
19	15, 16, 18	nf3an 1903	. . . . . . 7 ⊢ Ⅎ𝑓(𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥)
20	3, 19	nfxfr 1854	. . . . . 6 ⊢ Ⅎ𝑓𝜒
21		nfv 1916	. . . . . 6 ⊢ Ⅎ𝑓 𝑧 ∈ 𝐸
22	20, 21	nfan 1901	. . . . 5 ⊢ Ⅎ𝑓(𝜒 ∧ 𝑧 ∈ 𝐸)
23	2, 22	nfxfr 1854	. . . 4 ⊢ Ⅎ𝑓𝜃
24		nfv 1916	. . . 4 ⊢ Ⅎ𝑓 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)
25	23, 24	nfan 1901	. . 3 ⊢ Ⅎ𝑓(𝜃 ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅))
26	1, 25	nfxfr 1854	. 2 ⊢ Ⅎ𝑓𝜁
27	26	nf5ri 2187	1 ⊢ (𝜁 → ∀𝑓𝜁)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1086  ∀wal 1538   = wceq 1540  ∃wex 1780   ∈ wcel 2105  {cab 2708   ≠ wne 2939  ∀wral 3060  ∃wrex 3069  {crab 3431  [wsbc 3777   ∪ cun 3946   ⊆ wss 3948  ∅c0 4322  {csn 4628  ⟨cop 4634  ∪ cuni 4908   class class class wbr 5148  dom cdm 5676   ↾ cres 5678   Fn wfn 6538  ‘cfv 6543   predc-bnj14 33998   FrSe w-bnj15 34002   trClc-bnj18 34004

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rab 3432

This theorem is referenced by:  bnj1450  34360