Theorem bnj1452 31934

Description: Technical lemma for bnj60 31944. 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
bnj1452.1	⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1452.2	⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1452.3	⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
bnj1452.4	⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
bnj1452.5	⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}
bnj1452.6	⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))
bnj1452.7	⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))
bnj1452.8	⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)
bnj1452.9	⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
bnj1452.10	⊢ 𝑃 = ∪ 𝐻
bnj1452.11	⊢ 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1452.12	⊢ 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})
bnj1452.13	⊢ 𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩
bnj1452.14	⊢ 𝐸 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))

Assertion

Ref	Expression
bnj1452	⊢ (𝜒 → 𝐸 ∈ 𝐵)

Distinct variable groups:   𝐴,𝑑,𝑥,𝑧   𝐸,𝑑,𝑧   𝑅,𝑑,𝑥,𝑧   𝜒,𝑧

Allowed substitution hints:   𝜓(𝑥,𝑦,𝑧,𝑓,𝑑)   𝜒(𝑥,𝑦,𝑓,𝑑)   𝜏(𝑥,𝑦,𝑧,𝑓,𝑑)   𝐴(𝑦,𝑓)   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(𝑦,𝑓)   𝐸(𝑥,𝑦,𝑓)   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 bnj1452

Step	Hyp	Ref	Expression
1		bnj1452.14	. . 3 ⊢ 𝐸 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
2		bnj1452.5	. . . . . 6 ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}
3		bnj1452.7	. . . . . 6 ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))
4	2, 3	bnj1212 31684	. . . . 5 ⊢ (𝜒 → 𝑥 ∈ 𝐴)
5	4	snssd 4655	. . . 4 ⊢ (𝜒 → {𝑥} ⊆ 𝐴)
6		bnj1147 31876	. . . . 5 ⊢ trCl(𝑥, 𝐴, 𝑅) ⊆ 𝐴
7	6	a1i 11	. . . 4 ⊢ (𝜒 → trCl(𝑥, 𝐴, 𝑅) ⊆ 𝐴)
8	5, 7	unssd 4089	. . 3 ⊢ (𝜒 → ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) ⊆ 𝐴)
9	1, 8	eqsstrid 3942	. 2 ⊢ (𝜒 → 𝐸 ⊆ 𝐴)
10		elsni 4495	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑥} → 𝑧 = 𝑥)
11	10	adantl 482	. . . . . . 7 ⊢ (((𝜒 ∧ 𝑧 ∈ 𝐸) ∧ 𝑧 ∈ {𝑥}) → 𝑧 = 𝑥)
12		bnj602 31799	. . . . . . 7 ⊢ (𝑧 = 𝑥 → pred(𝑧, 𝐴, 𝑅) = pred(𝑥, 𝐴, 𝑅))
13	11, 12	syl 17	. . . . . 6 ⊢ (((𝜒 ∧ 𝑧 ∈ 𝐸) ∧ 𝑧 ∈ {𝑥}) → pred(𝑧, 𝐴, 𝑅) = pred(𝑥, 𝐴, 𝑅))
14		bnj1452.6	. . . . . . . . . 10 ⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))
15	14	simplbi 498	. . . . . . . . 9 ⊢ (𝜓 → 𝑅 FrSe 𝐴)
16	3, 15	bnj835 31643	. . . . . . . 8 ⊢ (𝜒 → 𝑅 FrSe 𝐴)
17		bnj906 31814	. . . . . . . 8 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → pred(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
18	16, 4, 17	syl2anc 584	. . . . . . 7 ⊢ (𝜒 → pred(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
19	18	ad2antrr 722	. . . . . 6 ⊢ (((𝜒 ∧ 𝑧 ∈ 𝐸) ∧ 𝑧 ∈ {𝑥}) → pred(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
20	13, 19	eqsstrd 3932	. . . . 5 ⊢ (((𝜒 ∧ 𝑧 ∈ 𝐸) ∧ 𝑧 ∈ {𝑥}) → pred(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
21		ssun4 4078	. . . . . 6 ⊢ ( pred(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅) → pred(𝑧, 𝐴, 𝑅) ⊆ ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
22	21, 1	syl6sseqr 3945	. . . . 5 ⊢ ( pred(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅) → pred(𝑧, 𝐴, 𝑅) ⊆ 𝐸)
23	20, 22	syl 17	. . . 4 ⊢ (((𝜒 ∧ 𝑧 ∈ 𝐸) ∧ 𝑧 ∈ {𝑥}) → pred(𝑧, 𝐴, 𝑅) ⊆ 𝐸)
24	16	ad2antrr 722	. . . . . . 7 ⊢ (((𝜒 ∧ 𝑧 ∈ 𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) → 𝑅 FrSe 𝐴)
25		simpr 485	. . . . . . . 8 ⊢ (((𝜒 ∧ 𝑧 ∈ 𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) → 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅))
26	6, 25	bnj1213 31683	. . . . . . 7 ⊢ (((𝜒 ∧ 𝑧 ∈ 𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) → 𝑧 ∈ 𝐴)
27		bnj906 31814	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑧 ∈ 𝐴) → pred(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑧, 𝐴, 𝑅))
28	24, 26, 27	syl2anc 584	. . . . . 6 ⊢ (((𝜒 ∧ 𝑧 ∈ 𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) → pred(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑧, 𝐴, 𝑅))
29	4	ad2antrr 722	. . . . . . 7 ⊢ (((𝜒 ∧ 𝑧 ∈ 𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) → 𝑥 ∈ 𝐴)
30		bnj1125 31874	. . . . . . 7 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) → trCl(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
31	24, 29, 25, 30	syl3anc 1364	. . . . . 6 ⊢ (((𝜒 ∧ 𝑧 ∈ 𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) → trCl(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
32	28, 31	sstrd 3905	. . . . 5 ⊢ (((𝜒 ∧ 𝑧 ∈ 𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) → pred(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
33	32, 22	syl 17	. . . 4 ⊢ (((𝜒 ∧ 𝑧 ∈ 𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) → pred(𝑧, 𝐴, 𝑅) ⊆ 𝐸)
34	1	bnj1424 31723	. . . . 5 ⊢ (𝑧 ∈ 𝐸 → (𝑧 ∈ {𝑥} ∨ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)))
35	34	adantl 482	. . . 4 ⊢ ((𝜒 ∧ 𝑧 ∈ 𝐸) → (𝑧 ∈ {𝑥} ∨ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)))
36	23, 33, 35	mpjaodan 953	. . 3 ⊢ ((𝜒 ∧ 𝑧 ∈ 𝐸) → pred(𝑧, 𝐴, 𝑅) ⊆ 𝐸)
37	36	ralrimiva 3151	. 2 ⊢ (𝜒 → ∀𝑧 ∈ 𝐸 pred(𝑧, 𝐴, 𝑅) ⊆ 𝐸)
38		snex 5230	. . . . . . . 8 ⊢ {𝑥} ∈ V
39	38	a1i 11	. . . . . . 7 ⊢ (𝜒 → {𝑥} ∈ V)
40		bnj893 31812	. . . . . . . 8 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → trCl(𝑥, 𝐴, 𝑅) ∈ V)
41	16, 4, 40	syl2anc 584	. . . . . . 7 ⊢ (𝜒 → trCl(𝑥, 𝐴, 𝑅) ∈ V)
42	39, 41	bnj1149 31677	. . . . . 6 ⊢ (𝜒 → ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) ∈ V)
43	1, 42	syl5eqel 2889	. . . . 5 ⊢ (𝜒 → 𝐸 ∈ V)
44		bnj1452.1	. . . . . 6 ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
45	44	bnj1454 31726	. . . . 5 ⊢ (𝐸 ∈ V → (𝐸 ∈ 𝐵 ↔ [𝐸 / 𝑑](𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)))
46	43, 45	syl 17	. . . 4 ⊢ (𝜒 → (𝐸 ∈ 𝐵 ↔ [𝐸 / 𝑑](𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)))
47		bnj602 31799	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → pred(𝑥, 𝐴, 𝑅) = pred(𝑧, 𝐴, 𝑅))
48	47	sseq1d 3925	. . . . . . 7 ⊢ (𝑥 = 𝑧 → ( pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑 ↔ pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑))
49	48	cbvralv 3405	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑 ↔ ∀𝑧 ∈ 𝑑 pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑)
50	49	anbi2i 622	. . . . 5 ⊢ ((𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑) ↔ (𝑑 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑑 pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑))
51	50	sbcbii 3763	. . . 4 ⊢ ([𝐸 / 𝑑](𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑) ↔ [𝐸 / 𝑑](𝑑 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑑 pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑))
52	46, 51	syl6bb 288	. . 3 ⊢ (𝜒 → (𝐸 ∈ 𝐵 ↔ [𝐸 / 𝑑](𝑑 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑑 pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑)))
53		sseq1 3919	. . . . . 6 ⊢ (𝑑 = 𝐸 → (𝑑 ⊆ 𝐴 ↔ 𝐸 ⊆ 𝐴))
54		sseq2 3920	. . . . . . 7 ⊢ (𝑑 = 𝐸 → ( pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑 ↔ pred(𝑧, 𝐴, 𝑅) ⊆ 𝐸))
55	54	raleqbi1dv 3365	. . . . . 6 ⊢ (𝑑 = 𝐸 → (∀𝑧 ∈ 𝑑 pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑 ↔ ∀𝑧 ∈ 𝐸 pred(𝑧, 𝐴, 𝑅) ⊆ 𝐸))
56	53, 55	anbi12d 630	. . . . 5 ⊢ (𝑑 = 𝐸 → ((𝑑 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑑 pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑) ↔ (𝐸 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐸 pred(𝑧, 𝐴, 𝑅) ⊆ 𝐸)))
57	56	sbcieg 3744	. . . 4 ⊢ (𝐸 ∈ V → ([𝐸 / 𝑑](𝑑 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑑 pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑) ↔ (𝐸 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐸 pred(𝑧, 𝐴, 𝑅) ⊆ 𝐸)))
58	43, 57	syl 17	. . 3 ⊢ (𝜒 → ([𝐸 / 𝑑](𝑑 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑑 pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑) ↔ (𝐸 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐸 pred(𝑧, 𝐴, 𝑅) ⊆ 𝐸)))
59	52, 58	bitrd 280	. 2 ⊢ (𝜒 → (𝐸 ∈ 𝐵 ↔ (𝐸 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐸 pred(𝑧, 𝐴, 𝑅) ⊆ 𝐸)))
60	9, 37, 59	mpbir2and 709	1 ⊢ (𝜒 → 𝐸 ∈ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 842   ∧ w3a 1080   = wceq 1525  ∃wex 1765   ∈ wcel 2083  {cab 2777   ≠ wne 2986  ∀wral 3107  ∃wrex 3108  {crab 3111  Vcvv 3440  [wsbc 3711   ∪ cun 3863   ⊆ wss 3865  ∅c0 4217  {csn 4478  ⟨cop 4484  ∪ cuni 4751   class class class wbr 4968  dom cdm 5450   ↾ cres 5452   Fn wfn 6227  ‘cfv 6232   predc-bnj14 31571   FrSe w-bnj15 31575   trClc-bnj18 31577

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1781  ax-4 1795  ax-5 1892  ax-6 1951  ax-7 1996  ax-8 2085  ax-9 2093  ax-10 2114  ax-11 2128  ax-12 2143  ax-13 2346  ax-ext 2771  ax-rep 5088  ax-sep 5101  ax-nul 5108  ax-pow 5164  ax-pr 5228  ax-un 7326  ax-reg 8909  ax-inf2 8957

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3or 1081  df-3an 1082  df-tru 1528  df-fal 1538  df-ex 1766  df-nf 1770  df-sb 2045  df-mo 2578  df-eu 2614  df-clab 2778  df-cleq 2790  df-clel 2865  df-nfc 2937  df-ne 2987  df-ral 3112  df-rex 3113  df-reu 3114  df-rab 3116  df-v 3442  df-sbc 3712  df-csb 3818  df-dif 3868  df-un 3870  df-in 3872  df-ss 3880  df-pss 3882  df-nul 4218  df-if 4388  df-pw 4461  df-sn 4479  df-pr 4481  df-tp 4483  df-op 4485  df-uni 4752  df-iun 4833  df-br 4969  df-opab 5031  df-mpt 5048  df-tr 5071  df-id 5355  df-eprel 5360  df-po 5369  df-so 5370  df-fr 5409  df-we 5411  df-xp 5456  df-rel 5457  df-cnv 5458  df-co 5459  df-dm 5460  df-rn 5461  df-res 5462  df-ima 5463  df-ord 6076  df-on 6077  df-lim 6078  df-suc 6079  df-iota 6196  df-fun 6234  df-fn 6235  df-f 6236  df-f1 6237  df-fo 6238  df-f1o 6239  df-fv 6240  df-om 7444  df-1o 7960  df-bnj17 31570  df-bnj14 31572  df-bnj13 31574  df-bnj15 31576  df-bnj18 31578  df-bnj19 31580

This theorem is referenced by:  bnj1312  31940