Theorem bnj1444 32923

Description: Technical lemma for bnj60 32942. 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
bnj1444.1	⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1444.2	⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1444.3	⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
bnj1444.4	⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
bnj1444.5	⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}
bnj1444.6	⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))
bnj1444.7	⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))
bnj1444.8	⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)
bnj1444.9	⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
bnj1444.10	⊢ 𝑃 = ∪ 𝐻
bnj1444.11	⊢ 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1444.12	⊢ 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})
bnj1444.13	⊢ 𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩
bnj1444.14	⊢ 𝐸 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
bnj1444.15	⊢ (𝜒 → 𝑃 Fn trCl(𝑥, 𝐴, 𝑅))
bnj1444.16	⊢ (𝜒 → 𝑄 Fn ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
bnj1444.17	⊢ (𝜃 ↔ (𝜒 ∧ 𝑧 ∈ 𝐸))
bnj1444.18	⊢ (𝜂 ↔ (𝜃 ∧ 𝑧 ∈ {𝑥}))
bnj1444.19	⊢ (𝜁 ↔ (𝜃 ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)))
bnj1444.20	⊢ (𝜌 ↔ (𝜁 ∧ 𝑓 ∈ 𝐻 ∧ 𝑧 ∈ dom 𝑓))

Assertion

Ref	Expression
bnj1444	⊢ (𝜌 → ∀𝑦𝜌)

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

Allowed substitution hints:   𝜓(𝑥,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)   𝜌(𝑥,𝑦,𝑧,𝑓,𝑑)   𝐴(𝑥,𝑧,𝑓,𝑑)   𝐵(𝑥,𝑦,𝑧,𝑓,𝑑)   𝐶(𝑥,𝑦,𝑧,𝑓,𝑑)   𝐷(𝑥,𝑧,𝑓,𝑑)   𝑃(𝑥,𝑦,𝑧,𝑓,𝑑)   𝑄(𝑥,𝑦,𝑧,𝑓,𝑑)   𝑅(𝑥,𝑧,𝑓,𝑑)   𝐸(𝑥,𝑧,𝑓,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 bnj1444

Step	Hyp	Ref	Expression
1		bnj1444.20	. . 3 ⊢ (𝜌 ↔ (𝜁 ∧ 𝑓 ∈ 𝐻 ∧ 𝑧 ∈ dom 𝑓))
2		bnj1444.19	. . . . 5 ⊢ (𝜁 ↔ (𝜃 ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)))
3		bnj1444.17	. . . . . . 7 ⊢ (𝜃 ↔ (𝜒 ∧ 𝑧 ∈ 𝐸))
4		bnj1444.7	. . . . . . . . 9 ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))
5		nfv 1918	. . . . . . . . . 10 ⊢ Ⅎ𝑦𝜓
6		nfv 1918	. . . . . . . . . 10 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐷
7		nfra1 3142	. . . . . . . . . 10 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥
8	5, 6, 7	nf3an 1905	. . . . . . . . 9 ⊢ Ⅎ𝑦(𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥)
9	4, 8	nfxfr 1856	. . . . . . . 8 ⊢ Ⅎ𝑦𝜒
10		nfv 1918	. . . . . . . 8 ⊢ Ⅎ𝑦 𝑧 ∈ 𝐸
11	9, 10	nfan 1903	. . . . . . 7 ⊢ Ⅎ𝑦(𝜒 ∧ 𝑧 ∈ 𝐸)
12	3, 11	nfxfr 1856	. . . . . 6 ⊢ Ⅎ𝑦𝜃
13		nfv 1918	. . . . . 6 ⊢ Ⅎ𝑦 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)
14	12, 13	nfan 1903	. . . . 5 ⊢ Ⅎ𝑦(𝜃 ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅))
15	2, 14	nfxfr 1856	. . . 4 ⊢ Ⅎ𝑦𝜁
16		bnj1444.9	. . . . . 6 ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
17		nfre1 3234	. . . . . . 7 ⊢ Ⅎ𝑦∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′
18	17	nfab 2912	. . . . . 6 ⊢ Ⅎ𝑦{𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
19	16, 18	nfcxfr 2904	. . . . 5 ⊢ Ⅎ𝑦𝐻
20	19	nfcri 2893	. . . 4 ⊢ Ⅎ𝑦 𝑓 ∈ 𝐻
21		nfv 1918	. . . 4 ⊢ Ⅎ𝑦 𝑧 ∈ dom 𝑓
22	15, 20, 21	nf3an 1905	. . 3 ⊢ Ⅎ𝑦(𝜁 ∧ 𝑓 ∈ 𝐻 ∧ 𝑧 ∈ dom 𝑓)
23	1, 22	nfxfr 1856	. 2 ⊢ Ⅎ𝑦𝜌
24	23	nf5ri 2191	1 ⊢ (𝜌 → ∀𝑦𝜌)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085  ∀wal 1537   = wceq 1539  ∃wex 1783   ∈ wcel 2108  {cab 2715   ≠ wne 2942  ∀wral 3063  ∃wrex 3064  {crab 3067  [wsbc 3711   ∪ cun 3881   ⊆ wss 3883  ∅c0 4253  {csn 4558  ⟨cop 4564  ∪ cuni 4836   class class class wbr 5070  dom cdm 5580   ↾ cres 5582   Fn wfn 6413  ‘cfv 6418   predc-bnj14 32567   FrSe w-bnj15 32571   trClc-bnj18 32573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069

This theorem is referenced by:  bnj1450  32930