Theorem bnj1446 32925

Description: Technical lemma for bnj60 32942. 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
bnj1446.1	⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1446.2	⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1446.3	⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
bnj1446.4	⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
bnj1446.5	⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}
bnj1446.6	⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))
bnj1446.7	⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))
bnj1446.8	⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)
bnj1446.9	⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
bnj1446.10	⊢ 𝑃 = ∪ 𝐻
bnj1446.11	⊢ 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1446.12	⊢ 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})
bnj1446.13	⊢ 𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩

Assertion

Ref	Expression
bnj1446	⊢ ((𝑄‘𝑧) = (𝐺‘𝑊) → ∀𝑑(𝑄‘𝑧) = (𝐺‘𝑊))

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

Allowed substitution hints:   𝜓(𝑥,𝑦,𝑧,𝑓,𝑑)   𝜒(𝑥,𝑦,𝑧,𝑓,𝑑)   𝜏(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)   𝐺(𝑥,𝑦,𝑧,𝑓)   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 bnj1446

Step	Hyp	Ref	Expression
1		bnj1446.12	. . . . 5 ⊢ 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})
2		bnj1446.10	. . . . . . 7 ⊢ 𝑃 = ∪ 𝐻
3		bnj1446.9	. . . . . . . . 9 ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
4		nfcv 2906	. . . . . . . . . . 11 ⊢ Ⅎ𝑑 pred(𝑥, 𝐴, 𝑅)
5		bnj1446.8	. . . . . . . . . . . 12 ⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)
6		nfcv 2906	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑑𝑦
7		bnj1446.4	. . . . . . . . . . . . . 14 ⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
8		bnj1446.3	. . . . . . . . . . . . . . . . 17 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
9		nfre1 3234	. . . . . . . . . . . . . . . . . 18 ⊢ Ⅎ𝑑∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))
10	9	nfab 2912	. . . . . . . . . . . . . . . . 17 ⊢ Ⅎ𝑑{𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
11	8, 10	nfcxfr 2904	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑑𝐶
12	11	nfcri 2893	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑑 𝑓 ∈ 𝐶
13		nfv 1918	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑑dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
14	12, 13	nfan 1903	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑑(𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
15	7, 14	nfxfr 1856	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑑𝜏
16	6, 15	nfsbcw 3733	. . . . . . . . . . . 12 ⊢ Ⅎ𝑑[𝑦 / 𝑥]𝜏
17	5, 16	nfxfr 1856	. . . . . . . . . . 11 ⊢ Ⅎ𝑑𝜏′
18	4, 17	nfrex 3237	. . . . . . . . . 10 ⊢ Ⅎ𝑑∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′
19	18	nfab 2912	. . . . . . . . 9 ⊢ Ⅎ𝑑{𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
20	3, 19	nfcxfr 2904	. . . . . . . 8 ⊢ Ⅎ𝑑𝐻
21	20	nfuni 4843	. . . . . . 7 ⊢ Ⅎ𝑑∪ 𝐻
22	2, 21	nfcxfr 2904	. . . . . 6 ⊢ Ⅎ𝑑𝑃
23		nfcv 2906	. . . . . . . 8 ⊢ Ⅎ𝑑𝑥
24		nfcv 2906	. . . . . . . . 9 ⊢ Ⅎ𝑑𝐺
25		bnj1446.11	. . . . . . . . . 10 ⊢ 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
26	22, 4	nfres 5882	. . . . . . . . . . 11 ⊢ Ⅎ𝑑(𝑃 ↾ pred(𝑥, 𝐴, 𝑅))
27	23, 26	nfop 4817	. . . . . . . . . 10 ⊢ Ⅎ𝑑⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
28	25, 27	nfcxfr 2904	. . . . . . . . 9 ⊢ Ⅎ𝑑𝑍
29	24, 28	nffv 6766	. . . . . . . 8 ⊢ Ⅎ𝑑(𝐺‘𝑍)
30	23, 29	nfop 4817	. . . . . . 7 ⊢ Ⅎ𝑑⟨𝑥, (𝐺‘𝑍)⟩
31	30	nfsn 4640	. . . . . 6 ⊢ Ⅎ𝑑{⟨𝑥, (𝐺‘𝑍)⟩}
32	22, 31	nfun 4095	. . . . 5 ⊢ Ⅎ𝑑(𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})
33	1, 32	nfcxfr 2904	. . . 4 ⊢ Ⅎ𝑑𝑄
34		nfcv 2906	. . . 4 ⊢ Ⅎ𝑑𝑧
35	33, 34	nffv 6766	. . 3 ⊢ Ⅎ𝑑(𝑄‘𝑧)
36		bnj1446.13	. . . . 5 ⊢ 𝑊 = ⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩
37		nfcv 2906	. . . . . . 7 ⊢ Ⅎ𝑑 pred(𝑧, 𝐴, 𝑅)
38	33, 37	nfres 5882	. . . . . 6 ⊢ Ⅎ𝑑(𝑄 ↾ pred(𝑧, 𝐴, 𝑅))
39	34, 38	nfop 4817	. . . . 5 ⊢ Ⅎ𝑑⟨𝑧, (𝑄 ↾ pred(𝑧, 𝐴, 𝑅))⟩
40	36, 39	nfcxfr 2904	. . . 4 ⊢ Ⅎ𝑑𝑊
41	24, 40	nffv 6766	. . 3 ⊢ Ⅎ𝑑(𝐺‘𝑊)
42	35, 41	nfeq 2919	. 2 ⊢ Ⅎ𝑑(𝑄‘𝑧) = (𝐺‘𝑊)
43	42	nf5ri 2191	1 ⊢ ((𝑄‘𝑧) = (𝐺‘𝑊) → ∀𝑑(𝑄‘𝑧) = (𝐺‘𝑊))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085  ∀wal 1537   = wceq 1539  ∃wex 1783   ∈ wcel 2108  {cab 2715   ≠ wne 2942  ∀wral 3063  ∃wrex 3064  {crab 3067  [wsbc 3711   ∪ cun 3881   ⊆ wss 3883  ∅c0 4253  {csn 4558  ⟨cop 4564  ∪ cuni 4836   class class class wbr 5070  dom cdm 5580   ↾ cres 5582   Fn wfn 6413  ‘cfv 6418   predc-bnj14 32567   FrSe w-bnj15 32571   trClc-bnj18 32573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-xp 5586  df-res 5592  df-iota 6376  df-fv 6426

This theorem is referenced by:  bnj1450  32930  bnj1463  32935