Theorem bnj1452 33032

Description: Technical lemma for bnj60 33042. 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 32779	. . . . 5 ⊢ (𝜒 → 𝑥 ∈ 𝐴)
5	4	snssd 4742	. . . 4 ⊢ (𝜒 → {𝑥} ⊆ 𝐴)
6		bnj1147 32974	. . . . 5 ⊢ trCl(𝑥, 𝐴, 𝑅) ⊆ 𝐴
7	6	a1i 11	. . . 4 ⊢ (𝜒 → trCl(𝑥, 𝐴, 𝑅) ⊆ 𝐴)
8	5, 7	unssd 4120	. . 3 ⊢ (𝜒 → ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) ⊆ 𝐴)
9	1, 8	eqsstrid 3969	. 2 ⊢ (𝜒 → 𝐸 ⊆ 𝐴)
10		elsni 4578	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑥} → 𝑧 = 𝑥)
11	10	adantl 482	. . . . . . 7 ⊢ (((𝜒 ∧ 𝑧 ∈ 𝐸) ∧ 𝑧 ∈ {𝑥}) → 𝑧 = 𝑥)
12		bnj602 32895	. . . . . . 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 32739	. . . . . . . 8 ⊢ (𝜒 → 𝑅 FrSe 𝐴)
17		bnj906 32910	. . . . . . . 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 723	. . . . . 6 ⊢ (((𝜒 ∧ 𝑧 ∈ 𝐸) ∧ 𝑧 ∈ {𝑥}) → pred(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
20	13, 19	eqsstrd 3959	. . . . 5 ⊢ (((𝜒 ∧ 𝑧 ∈ 𝐸) ∧ 𝑧 ∈ {𝑥}) → pred(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
21		ssun4 4109	. . . . . 6 ⊢ ( pred(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅) → pred(𝑧, 𝐴, 𝑅) ⊆ ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
22	21, 1	sseqtrrdi 3972	. . . . 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 723	. . . . . . 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 32778	. . . . . . 7 ⊢ (((𝜒 ∧ 𝑧 ∈ 𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) → 𝑧 ∈ 𝐴)
27		bnj906 32910	. . . . . . 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 723	. . . . . . 7 ⊢ (((𝜒 ∧ 𝑧 ∈ 𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) → 𝑥 ∈ 𝐴)
30		bnj1125 32972	. . . . . . 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 1370	. . . . . 6 ⊢ (((𝜒 ∧ 𝑧 ∈ 𝐸) ∧ 𝑧 ∈ trCl(𝑥, 𝐴, 𝑅)) → trCl(𝑧, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
32	28, 31	sstrd 3931	. . . . 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 32818	. . . . 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 956	. . 3 ⊢ ((𝜒 ∧ 𝑧 ∈ 𝐸) → pred(𝑧, 𝐴, 𝑅) ⊆ 𝐸)
37	36	ralrimiva 3103	. 2 ⊢ (𝜒 → ∀𝑧 ∈ 𝐸 pred(𝑧, 𝐴, 𝑅) ⊆ 𝐸)
38		snex 5354	. . . . . . . 8 ⊢ {𝑥} ∈ V
39	38	a1i 11	. . . . . . 7 ⊢ (𝜒 → {𝑥} ∈ V)
40		bnj893 32908	. . . . . . . 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 32772	. . . . . 6 ⊢ (𝜒 → ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) ∈ V)
43	1, 42	eqeltrid 2843	. . . . 5 ⊢ (𝜒 → 𝐸 ∈ V)
44		bnj1452.1	. . . . . 6 ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
45	44	bnj1454 32822	. . . . 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 32895	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → pred(𝑥, 𝐴, 𝑅) = pred(𝑧, 𝐴, 𝑅))
48	47	sseq1d 3952	. . . . . . 7 ⊢ (𝑥 = 𝑧 → ( pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑 ↔ pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑))
49	48	cbvralvw 3383	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑 ↔ ∀𝑧 ∈ 𝑑 pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑)
50	49	anbi2i 623	. . . . 5 ⊢ ((𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑) ↔ (𝑑 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑑 pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑))
51	50	sbcbii 3776	. . . 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	bitrdi 287	. . 3 ⊢ (𝜒 → (𝐸 ∈ 𝐵 ↔ [𝐸 / 𝑑](𝑑 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑑 pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑)))
53		sseq1 3946	. . . . . 6 ⊢ (𝑑 = 𝐸 → (𝑑 ⊆ 𝐴 ↔ 𝐸 ⊆ 𝐴))
54		sseq2 3947	. . . . . . 7 ⊢ (𝑑 = 𝐸 → ( pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑 ↔ pred(𝑧, 𝐴, 𝑅) ⊆ 𝐸))
55	54	raleqbi1dv 3340	. . . . . 6 ⊢ (𝑑 = 𝐸 → (∀𝑧 ∈ 𝑑 pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑 ↔ ∀𝑧 ∈ 𝐸 pred(𝑧, 𝐴, 𝑅) ⊆ 𝐸))
56	53, 55	anbi12d 631	. . . . 5 ⊢ (𝑑 = 𝐸 → ((𝑑 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝑑 pred(𝑧, 𝐴, 𝑅) ⊆ 𝑑) ↔ (𝐸 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐸 pred(𝑧, 𝐴, 𝑅) ⊆ 𝐸)))
57	56	sbcieg 3756	. . . 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 278	. 2 ⊢ (𝜒 → (𝐸 ∈ 𝐵 ↔ (𝐸 ⊆ 𝐴 ∧ ∀𝑧 ∈ 𝐸 pred(𝑧, 𝐴, 𝑅) ⊆ 𝐸)))
60	9, 37, 59	mpbir2and 710	1 ⊢ (𝜒 → 𝐸 ∈ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844   ∧ w3a 1086   = wceq 1539  ∃wex 1782   ∈ wcel 2106  {cab 2715   ≠ wne 2943  ∀wral 3064  ∃wrex 3065  {crab 3068  Vcvv 3432  [wsbc 3716   ∪ cun 3885   ⊆ wss 3887  ∅c0 4256  {csn 4561  ⟨cop 4567  ∪ cuni 4839   class class class wbr 5074  dom cdm 5589   ↾ cres 5591   Fn wfn 6428  ‘cfv 6433   predc-bnj14 32667   FrSe w-bnj15 32671   trClc-bnj18 32673

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-reg 9351  ax-inf2 9399

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-om 7713  df-1o 8297  df-bnj17 32666  df-bnj14 32668  df-bnj13 32670  df-bnj15 32672  df-bnj18 32674  df-bnj19 32676

This theorem is referenced by:  bnj1312  33038