Theorem bnj1442 33750

Description: Technical lemma for bnj60 33763. 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
bnj1442.1	⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1442.2	⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1442.3	⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
bnj1442.4	⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
bnj1442.5	⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}
bnj1442.6	⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))
bnj1442.7	⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))
bnj1442.8	⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)
bnj1442.9	⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
bnj1442.10	⊢ 𝑃 = ∪ 𝐻
bnj1442.11	⊢ 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1442.12	⊢ 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})
bnj1442.13	⊢ 𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩
bnj1442.14	⊢ 𝐸 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
bnj1442.15	⊢ (𝜒 → 𝑃 Fn trCl(𝑥, 𝐴, 𝑅))
bnj1442.16	⊢ (𝜒 → 𝑄 Fn ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
bnj1442.17	⊢ (𝜃 ↔ (𝜒 ∧ 𝑧 ∈ 𝐸))
bnj1442.18	⊢ (𝜂 ↔ (𝜃 ∧ 𝑧 ∈ {𝑥}))

Assertion

Ref	Expression
bnj1442	⊢ (𝜂 → (𝑄‘𝑧) = (𝐺‘𝑊))

Distinct variable group:   𝑥,𝐴

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)   𝐴(𝑦,𝑧,𝑓,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,𝑓,𝑑)   𝐸(𝑥,𝑦,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 bnj1442

Step	Hyp	Ref	Expression
1		bnj1442.18	. . 3 ⊢ (𝜂 ↔ (𝜃 ∧ 𝑧 ∈ {𝑥}))
2		bnj1442.17	. . . 4 ⊢ (𝜃 ↔ (𝜒 ∧ 𝑧 ∈ 𝐸))
3		bnj1442.16	. . . . . 6 ⊢ (𝜒 → 𝑄 Fn ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
4	3	fnfund 6608	. . . . 5 ⊢ (𝜒 → Fun 𝑄)
5		opex 5426	. . . . . . . 8 ⊢ ⟨𝑥, (𝐺‘𝑍)⟩ ∈ V
6	5	snid 4627	. . . . . . 7 ⊢ ⟨𝑥, (𝐺‘𝑍)⟩ ∈ {⟨𝑥, (𝐺‘𝑍)⟩}
7		elun2 4142	. . . . . . 7 ⊢ (⟨𝑥, (𝐺‘𝑍)⟩ ∈ {⟨𝑥, (𝐺‘𝑍)⟩} → ⟨𝑥, (𝐺‘𝑍)⟩ ∈ (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩}))
8	6, 7	ax-mp 5	. . . . . 6 ⊢ ⟨𝑥, (𝐺‘𝑍)⟩ ∈ (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})
9		bnj1442.12	. . . . . 6 ⊢ 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})
10	8, 9	eleqtrri 2831	. . . . 5 ⊢ ⟨𝑥, (𝐺‘𝑍)⟩ ∈ 𝑄
11		funopfv 6899	. . . . 5 ⊢ (Fun 𝑄 → (⟨𝑥, (𝐺‘𝑍)⟩ ∈ 𝑄 → (𝑄‘𝑥) = (𝐺‘𝑍)))
12	4, 10, 11	mpisyl 21	. . . 4 ⊢ (𝜒 → (𝑄‘𝑥) = (𝐺‘𝑍))
13	2, 12	bnj832 33459	. . 3 ⊢ (𝜃 → (𝑄‘𝑥) = (𝐺‘𝑍))
14	1, 13	bnj832 33459	. 2 ⊢ (𝜂 → (𝑄‘𝑥) = (𝐺‘𝑍))
15		elsni 4608	. . . 4 ⊢ (𝑧 ∈ {𝑥} → 𝑧 = 𝑥)
16	1, 15	simplbiim 505	. . 3 ⊢ (𝜂 → 𝑧 = 𝑥)
17	16	fveq2d 6851	. 2 ⊢ (𝜂 → (𝑄‘𝑧) = (𝑄‘𝑥))
18		bnj602 33616	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → pred(𝑧, 𝐴, 𝑅) = pred(𝑥, 𝐴, 𝑅))
19	18	reseq2d 5942	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝑄 ↾ pred(𝑧, 𝐴, 𝑅)) = (𝑄 ↾ pred(𝑥, 𝐴, 𝑅)))
20	16, 19	syl 17	. . . . . 6 ⊢ (𝜂 → (𝑄 ↾ pred(𝑧, 𝐴, 𝑅)) = (𝑄 ↾ pred(𝑥, 𝐴, 𝑅)))
21	9	bnj931 33471	. . . . . . . . . 10 ⊢ 𝑃 ⊆ 𝑄
22	21	a1i 11	. . . . . . . . 9 ⊢ (𝜒 → 𝑃 ⊆ 𝑄)
23		bnj1442.7	. . . . . . . . . . . 12 ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))
24		bnj1442.6	. . . . . . . . . . . . 13 ⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))
25	24	simplbi 498	. . . . . . . . . . . 12 ⊢ (𝜓 → 𝑅 FrSe 𝐴)
26	23, 25	bnj835 33460	. . . . . . . . . . 11 ⊢ (𝜒 → 𝑅 FrSe 𝐴)
27		bnj1442.5	. . . . . . . . . . . 12 ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}
28	27, 23	bnj1212 33500	. . . . . . . . . . 11 ⊢ (𝜒 → 𝑥 ∈ 𝐴)
29		bnj906 33631	. . . . . . . . . . 11 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → pred(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
30	26, 28, 29	syl2anc 584	. . . . . . . . . 10 ⊢ (𝜒 → pred(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
31		bnj1442.15	. . . . . . . . . . 11 ⊢ (𝜒 → 𝑃 Fn trCl(𝑥, 𝐴, 𝑅))
32	31	fndmd 6612	. . . . . . . . . 10 ⊢ (𝜒 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅))
33	30, 32	sseqtrrd 3988	. . . . . . . . 9 ⊢ (𝜒 → pred(𝑥, 𝐴, 𝑅) ⊆ dom 𝑃)
34	4, 22, 33	bnj1503 33550	. . . . . . . 8 ⊢ (𝜒 → (𝑄 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑃 ↾ pred(𝑥, 𝐴, 𝑅)))
35	2, 34	bnj832 33459	. . . . . . 7 ⊢ (𝜃 → (𝑄 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑃 ↾ pred(𝑥, 𝐴, 𝑅)))
36	1, 35	bnj832 33459	. . . . . 6 ⊢ (𝜂 → (𝑄 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑃 ↾ pred(𝑥, 𝐴, 𝑅)))
37	20, 36	eqtrd 2771	. . . . 5 ⊢ (𝜂 → (𝑄 ↾ pred(𝑧, 𝐴, 𝑅)) = (𝑃 ↾ pred(𝑥, 𝐴, 𝑅)))
38	16, 37	opeq12d 4843	. . . 4 ⊢ (𝜂 → ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩ = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩)
39		bnj1442.13	. . . 4 ⊢ 𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩
40		bnj1442.11	. . . 4 ⊢ 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
41	38, 39, 40	3eqtr4g 2796	. . 3 ⊢ (𝜂 → 𝑊 = 𝑍)
42	41	fveq2d 6851	. 2 ⊢ (𝜂 → (𝐺‘𝑊) = (𝐺‘𝑍))
43	14, 17, 42	3eqtr4d 2781	1 ⊢ (𝜂 → (𝑄‘𝑧) = (𝐺‘𝑊))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1087   = wceq 1541  ∃wex 1781   ∈ wcel 2106  {cab 2708   ≠ wne 2939  ∀wral 3060  ∃wrex 3069  {crab 3405  [wsbc 3742   ∪ cun 3911   ⊆ wss 3913  ∅c0 4287  {csn 4591  ⟨cop 4597  ∪ cuni 4870   class class class wbr 5110  dom cdm 5638   ↾ cres 5640  Fun wfun 6495   Fn wfn 6496  ‘cfv 6501   predc-bnj14 33389   FrSe w-bnj15 33393   trClc-bnj18 33395

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2702  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-reg 9537  ax-inf2 9586

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-om 7808  df-1o 8417  df-bnj17 33388  df-bnj14 33390  df-bnj13 33392  df-bnj15 33394  df-bnj18 33396

This theorem is referenced by:  bnj1423  33752