Theorem bnj1442 35246

Description: Technical lemma for bnj60 35259. 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 6590	. . . . 5 ⊢ (𝜒 → Fun 𝑄)
5		opex 5406	. . . . . . . 8 ⊢ ⟨𝑥, (𝐺‘𝑍)⟩ ∈ V
6	5	snid 4597	. . . . . . 7 ⊢ ⟨𝑥, (𝐺‘𝑍)⟩ ∈ {⟨𝑥, (𝐺‘𝑍)⟩}
7		elun2 4115	. . . . . . 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 2840	. . . . 5 ⊢ ⟨𝑥, (𝐺‘𝑍)⟩ ∈ 𝑄
11		funopfv 6880	. . . . 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 34956	. . 3 ⊢ (𝜃 → (𝑄‘𝑥) = (𝐺‘𝑍))
14	1, 13	bnj832 34956	. 2 ⊢ (𝜂 → (𝑄‘𝑥) = (𝐺‘𝑍))
15		elsni 4575	. . . 4 ⊢ (𝑧 ∈ {𝑥} → 𝑧 = 𝑥)
16	1, 15	simplbiim 510	. . 3 ⊢ (𝜂 → 𝑧 = 𝑥)
17	16	fveq2d 6835	. 2 ⊢ (𝜂 → (𝑄‘𝑧) = (𝑄‘𝑥))
18		bnj602 35112	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → pred(𝑧, 𝐴, 𝑅) = pred(𝑥, 𝐴, 𝑅))
19	18	reseq2d 5938	. . . . . . 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 34968	. . . . . . . . . 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 34957	. . . . . . . . . . 11 ⊢ (𝜒 → 𝑅 FrSe 𝐴)
27		bnj1442.5	. . . . . . . . . . . 12 ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}
28	27, 23	bnj1212 34996	. . . . . . . . . . 11 ⊢ (𝜒 → 𝑥 ∈ 𝐴)
29		bnj906 35127	. . . . . . . . . . 11 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑥 ∈ 𝐴) → pred(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
30	26, 28, 29	syl2anc 591	. . . . . . . . . 10 ⊢ (𝜒 → pred(𝑥, 𝐴, 𝑅) ⊆ trCl(𝑥, 𝐴, 𝑅))
31		bnj1442.15	. . . . . . . . . . 11 ⊢ (𝜒 → 𝑃 Fn trCl(𝑥, 𝐴, 𝑅))
32	31	fndmd 6594	. . . . . . . . . 10 ⊢ (𝜒 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅))
33	30, 32	sseqtrrd 3954	. . . . . . . . 9 ⊢ (𝜒 → pred(𝑥, 𝐴, 𝑅) ⊆ dom 𝑃)
34	4, 22, 33	bnj1503 35046	. . . . . . . 8 ⊢ (𝜒 → (𝑄 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑃 ↾ pred(𝑥, 𝐴, 𝑅)))
35	2, 34	bnj832 34956	. . . . . . 7 ⊢ (𝜃 → (𝑄 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑃 ↾ pred(𝑥, 𝐴, 𝑅)))
36	1, 35	bnj832 34956	. . . . . 6 ⊢ (𝜂 → (𝑄 ↾ pred(𝑥, 𝐴, 𝑅)) = (𝑃 ↾ pred(𝑥, 𝐴, 𝑅)))
37	20, 36	eqtrd 2776	. . . . 5 ⊢ (𝜂 → (𝑄 ↾ pred(𝑧, 𝐴, 𝑅)) = (𝑃 ↾ pred(𝑥, 𝐴, 𝑅)))
38	16, 37	opeq12d 4815	. . . 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 2801	. . 3 ⊢ (𝜂 → 𝑊 = 𝑍)
42	41	fveq2d 6835	. 2 ⊢ (𝜂 → (𝐺‘𝑊) = (𝐺‘𝑍))
43	14, 17, 42	3eqtr4d 2786	1 ⊢ (𝜂 → (𝑄‘𝑧) = (𝐺‘𝑊))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 397   ∧ w3a 1093   = wceq 1548  ∃wex 1787   ∈ wcel 2121  {cab 2719   ≠ wne 2936  ∀wral 3055  ∃wrex 3065  {crab 3393  [wsbc 3725   ∪ cun 3883   ⊆ wss 3885  ∅c0 4264  {csn 4558  ⟨cop 4564  ∪ cuni 4841   class class class wbr 5075  dom cdm 5621   ↾ cres 5623  Fun wfun 6483   Fn wfn 6484  ‘cfv 6489   predc-bnj14 34886   FrSe w-bnj15 34890   trClc-bnj18 34892

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682  ax-reg 9501  ax-inf2 9557

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-om 7811  df-1o 8399  df-bnj17 34885  df-bnj14 34887  df-bnj13 34889  df-bnj15 34891  df-bnj18 34893

This theorem is referenced by:  bnj1423  35248