Theorem bnj1467 32321

Description: Technical lemma for bnj60 32329. 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
bnj1467.1	⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1467.2	⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1467.3	⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
bnj1467.4	⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
bnj1467.5	⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}
bnj1467.6	⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))
bnj1467.7	⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))
bnj1467.8	⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)
bnj1467.9	⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
bnj1467.10	⊢ 𝑃 = ∪ 𝐻
bnj1467.11	⊢ 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1467.12	⊢ 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})

Assertion

Ref	Expression
bnj1467	⊢ (𝑤 ∈ 𝑄 → ∀𝑑 𝑤 ∈ 𝑄)

Distinct variable groups:   𝐴,𝑑,𝑤,𝑥   𝐵,𝑓   𝑤,𝐶   𝐺,𝑑,𝑤   𝑤,𝐻   𝑤,𝑃   𝑅,𝑑,𝑤,𝑥   𝑤,𝑍   𝑓,𝑑,𝑤,𝑥   𝑦,𝑑,𝑥

Allowed substitution hints:   𝜓(𝑥,𝑦,𝑤,𝑓,𝑑)   𝜒(𝑥,𝑦,𝑤,𝑓,𝑑)   𝜏(𝑥,𝑦,𝑤,𝑓,𝑑)   𝐴(𝑦,𝑓)   𝐵(𝑥,𝑦,𝑤,𝑑)   𝐶(𝑥,𝑦,𝑓,𝑑)   𝐷(𝑥,𝑦,𝑤,𝑓,𝑑)   𝑃(𝑥,𝑦,𝑓,𝑑)   𝑄(𝑥,𝑦,𝑤,𝑓,𝑑)   𝑅(𝑦,𝑓)   𝐺(𝑥,𝑦,𝑓)   𝐻(𝑥,𝑦,𝑓,𝑑)   𝑌(𝑥,𝑦,𝑤,𝑓,𝑑)   𝑍(𝑥,𝑦,𝑓,𝑑)   𝜏′(𝑥,𝑦,𝑤,𝑓,𝑑)

Proof of Theorem bnj1467

Step	Hyp	Ref	Expression
1		bnj1467.12	. . 3 ⊢ 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})
2		bnj1467.10	. . . . 5 ⊢ 𝑃 = ∪ 𝐻
3		bnj1467.9	. . . . . . 7 ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
4		nfcv 2977	. . . . . . . . 9 ⊢ Ⅎ𝑑 pred(𝑥, 𝐴, 𝑅)
5		bnj1467.8	. . . . . . . . . 10 ⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)
6		nfcv 2977	. . . . . . . . . . 11 ⊢ Ⅎ𝑑𝑦
7		bnj1467.4	. . . . . . . . . . . 12 ⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
8		bnj1467.3	. . . . . . . . . . . . . . 15 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
9		nfre1 3306	. . . . . . . . . . . . . . . 16 ⊢ Ⅎ𝑑∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))
10	9	nfab 2984	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑑{𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
11	8, 10	nfcxfr 2975	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑑𝐶
12	11	nfcri 2971	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑑 𝑓 ∈ 𝐶
13		nfv 1911	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑑dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
14	12, 13	nfan 1896	. . . . . . . . . . . 12 ⊢ Ⅎ𝑑(𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
15	7, 14	nfxfr 1849	. . . . . . . . . . 11 ⊢ Ⅎ𝑑𝜏
16	6, 15	nfsbcw 3793	. . . . . . . . . 10 ⊢ Ⅎ𝑑[𝑦 / 𝑥]𝜏
17	5, 16	nfxfr 1849	. . . . . . . . 9 ⊢ Ⅎ𝑑𝜏′
18	4, 17	nfrex 3309	. . . . . . . 8 ⊢ Ⅎ𝑑∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′
19	18	nfab 2984	. . . . . . 7 ⊢ Ⅎ𝑑{𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
20	3, 19	nfcxfr 2975	. . . . . 6 ⊢ Ⅎ𝑑𝐻
21	20	nfuni 4838	. . . . 5 ⊢ Ⅎ𝑑∪ 𝐻
22	2, 21	nfcxfr 2975	. . . 4 ⊢ Ⅎ𝑑𝑃
23		nfcv 2977	. . . . . 6 ⊢ Ⅎ𝑑𝑥
24		nfcv 2977	. . . . . . 7 ⊢ Ⅎ𝑑𝐺
25		bnj1467.11	. . . . . . . 8 ⊢ 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
26	22, 4	nfres 5849	. . . . . . . . 9 ⊢ Ⅎ𝑑(𝑃 ↾ pred(𝑥, 𝐴, 𝑅))
27	23, 26	nfop 4812	. . . . . . . 8 ⊢ Ⅎ𝑑⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
28	25, 27	nfcxfr 2975	. . . . . . 7 ⊢ Ⅎ𝑑𝑍
29	24, 28	nffv 6674	. . . . . 6 ⊢ Ⅎ𝑑(𝐺‘𝑍)
30	23, 29	nfop 4812	. . . . 5 ⊢ Ⅎ𝑑⟨𝑥, (𝐺‘𝑍)⟩
31	30	nfsn 4636	. . . 4 ⊢ Ⅎ𝑑{⟨𝑥, (𝐺‘𝑍)⟩}
32	22, 31	nfun 4140	. . 3 ⊢ Ⅎ𝑑(𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})
33	1, 32	nfcxfr 2975	. 2 ⊢ Ⅎ𝑑𝑄
34	33	nfcrii 2970	1 ⊢ (𝑤 ∈ 𝑄 → ∀𝑑 𝑤 ∈ 𝑄)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083  ∀wal 1531   = wceq 1533  ∃wex 1776   ∈ wcel 2110  {cab 2799   ≠ wne 3016  ∀wral 3138  ∃wrex 3139  {crab 3142  [wsbc 3771   ∪ cun 3933   ⊆ wss 3935  ∅c0 4290  {csn 4560  ⟨cop 4566  ∪ cuni 4831   class class class wbr 5058  dom cdm 5549   ↾ cres 5551   Fn wfn 6344  ‘cfv 6349   predc-bnj14 31953   FrSe w-bnj15 31957   trClc-bnj18 31959

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-xp 5555  df-res 5561  df-iota 6308  df-fv 6357

This theorem is referenced by:  bnj1463  32322