Theorem bnj1445 30366

Description: Technical lemma for bnj60 30384. 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
bnj1445.1	⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1445.2	⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1445.3	⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
bnj1445.4	⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
bnj1445.5	⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}
bnj1445.6	⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))
bnj1445.7	⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))
bnj1445.8	⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)
bnj1445.9	⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
bnj1445.10	⊢ 𝑃 = ∪ 𝐻
bnj1445.11	⊢ 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1445.12	⊢ 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})
bnj1445.13	⊢ 𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩
bnj1445.14	⊢ 𝐸 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
bnj1445.15	⊢ (𝜒 → 𝑃 Fn trCl(𝑥, 𝐴, 𝑅))
bnj1445.16	⊢ (𝜒 → 𝑄 Fn ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
bnj1445.17	⊢ (𝜃 ↔ (𝜒 ∧ 𝑧 ∈ 𝐸))
bnj1445.18	⊢ (𝜂 ↔ (𝜃 ∧ 𝑧 ∈ {𝑥}))
bnj1445.19	⊢ (𝜁 ↔ (𝜃 ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)))
bnj1445.20	⊢ (𝜌 ↔ (𝜁 ∧ 𝑓 ∈ 𝐻 ∧ 𝑧 ∈ dom 𝑓))
bnj1445.21	⊢ (𝜎 ↔ (𝜌 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
bnj1445.22	⊢ (𝜑 ↔ (𝜎 ∧ 𝑑 ∈ 𝐵 ∧ 𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)))
bnj1445.23	⊢ 𝑋 = ⟨𝑧, (𝑓 ↾ pred(𝑧, 𝐴, 𝑅))⟩

Assertion

Ref	Expression
bnj1445	⊢ (𝜎 → ∀𝑑𝜎)

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

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧,𝑓,𝑑)   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)   𝜎(𝑥,𝑦,𝑧,𝑓,𝑑)   𝜌(𝑥,𝑦,𝑧,𝑓,𝑑)   𝐴(𝑦,𝑧,𝑓)   𝐵(𝑥,𝑦,𝑧,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(𝑦,𝑧,𝑓)   𝐸(𝑥,𝑦,𝑧,𝑓)   𝐺(𝑥,𝑦,𝑧,𝑓,𝑑)   𝐻(𝑥,𝑦,𝑧,𝑓,𝑑)   𝑊(𝑥,𝑦,𝑧,𝑓,𝑑)   𝑋(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 bnj1445

Step	Hyp	Ref	Expression
1		bnj1445.21	. 2 ⊢ (𝜎 ↔ (𝜌 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
2		bnj1445.20	. . . . 5 ⊢ (𝜌 ↔ (𝜁 ∧ 𝑓 ∈ 𝐻 ∧ 𝑧 ∈ dom 𝑓))
3		bnj1445.19	. . . . . . 7 ⊢ (𝜁 ↔ (𝜃 ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)))
4		bnj1445.17	. . . . . . . . 9 ⊢ (𝜃 ↔ (𝜒 ∧ 𝑧 ∈ 𝐸))
5		bnj1445.7	. . . . . . . . . . . . 13 ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))
6		bnj1445.6	. . . . . . . . . . . . . . 15 ⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))
7		nfv 1830	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑑 𝑅 FrSe 𝐴
8		bnj1445.5	. . . . . . . . . . . . . . . . . 18 ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}
9		bnj1445.4	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
10		bnj1445.3	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
11		nfre1 2988	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ Ⅎ𝑑∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))
12	11	nfab 2755	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ Ⅎ𝑑{𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
13	10, 12	nfcxfr 2749	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ Ⅎ𝑑𝐶
14	13	nfcri 2745	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑑 𝑓 ∈ 𝐶
15		nfv 1830	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ Ⅎ𝑑dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
16	14, 15	nfan 1816	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ Ⅎ𝑑(𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
17	9, 16	nfxfr 1771	. . . . . . . . . . . . . . . . . . . . 21 ⊢ Ⅎ𝑑𝜏
18	17	nfex 2140	. . . . . . . . . . . . . . . . . . . 20 ⊢ Ⅎ𝑑∃𝑓𝜏
19	18	nfn 1768	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑑 ¬ ∃𝑓𝜏
20		nfcv 2751	. . . . . . . . . . . . . . . . . . 19 ⊢ Ⅎ𝑑𝐴
21	19, 20	nfrab 3100	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑑{𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}
22	8, 21	nfcxfr 2749	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑑𝐷
23		nfcv 2751	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑑∅
24	22, 23	nfne 2882	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑑 𝐷 ≠ ∅
25	7, 24	nfan 1816	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑑(𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅)
26	6, 25	nfxfr 1771	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑑𝜓
27	22	nfcri 2745	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑑 𝑥 ∈ 𝐷
28		nfv 1830	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑑 ¬ 𝑦𝑅𝑥
29	22, 28	nfral 2929	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑑∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥
30	26, 27, 29	nf3an 1819	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑑(𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥)
31	5, 30	nfxfr 1771	. . . . . . . . . . . 12 ⊢ Ⅎ𝑑𝜒
32	31	nf5ri 2053	. . . . . . . . . . 11 ⊢ (𝜒 → ∀𝑑𝜒)
33	32	bnj1351 30151	. . . . . . . . . 10 ⊢ ((𝜒 ∧ 𝑧 ∈ 𝐸) → ∀𝑑(𝜒 ∧ 𝑧 ∈ 𝐸))
34	33	nf5i 2011	. . . . . . . . 9 ⊢ Ⅎ𝑑(𝜒 ∧ 𝑧 ∈ 𝐸)
35	4, 34	nfxfr 1771	. . . . . . . 8 ⊢ Ⅎ𝑑𝜃
36		nfv 1830	. . . . . . . 8 ⊢ Ⅎ𝑑 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)
37	35, 36	nfan 1816	. . . . . . 7 ⊢ Ⅎ𝑑(𝜃 ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅))
38	3, 37	nfxfr 1771	. . . . . 6 ⊢ Ⅎ𝑑𝜁
39		bnj1445.9	. . . . . . . 8 ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
40		nfcv 2751	. . . . . . . . . 10 ⊢ Ⅎ𝑑 pred(𝑥, 𝐴, 𝑅)
41		bnj1445.8	. . . . . . . . . . 11 ⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)
42		nfcv 2751	. . . . . . . . . . . 12 ⊢ Ⅎ𝑑𝑦
43	42, 17	nfsbc 3424	. . . . . . . . . . 11 ⊢ Ⅎ𝑑[𝑦 / 𝑥]𝜏
44	41, 43	nfxfr 1771	. . . . . . . . . 10 ⊢ Ⅎ𝑑𝜏′
45	40, 44	nfrex 2990	. . . . . . . . 9 ⊢ Ⅎ𝑑∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′
46	45	nfab 2755	. . . . . . . 8 ⊢ Ⅎ𝑑{𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
47	39, 46	nfcxfr 2749	. . . . . . 7 ⊢ Ⅎ𝑑𝐻
48	47	nfcri 2745	. . . . . 6 ⊢ Ⅎ𝑑 𝑓 ∈ 𝐻
49		nfv 1830	. . . . . 6 ⊢ Ⅎ𝑑 𝑧 ∈ dom 𝑓
50	38, 48, 49	nf3an 1819	. . . . 5 ⊢ Ⅎ𝑑(𝜁 ∧ 𝑓 ∈ 𝐻 ∧ 𝑧 ∈ dom 𝑓)
51	2, 50	nfxfr 1771	. . . 4 ⊢ Ⅎ𝑑𝜌
52	51	nf5ri 2053	. . 3 ⊢ (𝜌 → ∀𝑑𝜌)
53		ax-5 1827	. . 3 ⊢ (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) → ∀𝑑 𝑦 ∈ pred(𝑥, 𝐴, 𝑅))
54	14	nf5ri 2053	. . 3 ⊢ (𝑓 ∈ 𝐶 → ∀𝑑 𝑓 ∈ 𝐶)
55		ax-5 1827	. . 3 ⊢ (dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)) → ∀𝑑dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
56	52, 53, 54, 55	bnj982 30103	. 2 ⊢ ((𝜌 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))) → ∀𝑑(𝜌 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
57	1, 56	hbxfrbi 1742	1 ⊢ (𝜎 → ∀𝑑𝜎)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  {crab 2900  [wsbc 3402   ∪ cun 3538   ⊆ wss 3540  ∅c0 3874  {csn 4125  ⟨cop 4131  ∪ cuni 4372   class class class wbr 4583  dom cdm 5038   ↾ cres 5040   Fn wfn 5799  ‘cfv 5804   ∧ w-bnj17 30005   predc-bnj14 30007   FrSe w-bnj15 30011   trClc-bnj18 30013

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-sbc 3403  df-bnj17 30006

This theorem is referenced by:  bnj1450  30372