Theorem bnj1466 35683

Description: Technical lemma for bnj60 35692. 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
bnj1466.1	⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1466.2	⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1466.3	⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
bnj1466.4	⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
bnj1466.5	⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}
bnj1466.6	⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))
bnj1466.7	⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))
bnj1466.8	⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)
bnj1466.9	⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
bnj1466.10	⊢ 𝑃 = ∪ 𝐻
bnj1466.11	⊢ 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1466.12	⊢ 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})

Assertion

Ref	Expression
bnj1466	⊢ (𝑤 ∈ 𝑄 → ∀𝑓 𝑤 ∈ 𝑄)

Distinct variable groups:   𝐴,𝑓,𝑤   𝑓,𝐺,𝑤   𝑤,𝐻   𝑤,𝑃   𝑅,𝑓,𝑤   𝑤,𝑍   𝑥,𝑓,𝑤

Allowed substitution hints:   𝜓(𝑥, 𝑦, 𝑤, 𝑓, 𝑑)   𝜒(𝑥, 𝑦, 𝑤, 𝑓, 𝑑)   𝜏(𝑥, 𝑦, 𝑤, 𝑓, 𝑑)   𝐴(𝑥, 𝑦, 𝑑)   𝐵(𝑥, 𝑦, 𝑤, 𝑓, 𝑑)   𝐶(𝑥, 𝑦, 𝑤, 𝑓, 𝑑)   𝐷(𝑥, 𝑦, 𝑤, 𝑓, 𝑑)   𝑃(𝑥, 𝑦, 𝑓, 𝑑)   𝑄(𝑥, 𝑦, 𝑤, 𝑓, 𝑑)   𝑅(𝑥, 𝑦, 𝑑)   𝐺(𝑥, 𝑦, 𝑑)   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 bnj1466

Step	Hyp	Ref	Expression
1		bnj1466.12	. . 3 ⊢ 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})
2		bnj1466.10	. . . . 5 ⊢ 𝑃 = ∪ 𝐻
3		bnj1466.9	. . . . . . . 8 ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
4	3	bnj1317 35451	. . . . . . 7 ⊢ (𝑤 ∈ 𝐻 → ∀𝑓 𝑤 ∈ 𝐻)
5	4	nfcii 2911	. . . . . 6 ⊢ Ⅎ𝑓𝐻
6	5	nfuni 4873	. . . . 5 ⊢ Ⅎ𝑓∪ 𝐻
7	2, 6	nfcxfr 2920	. . . 4 ⊢ Ⅎ𝑓𝑃
8		nfcv 2922	. . . . . 6 ⊢ Ⅎ𝑓𝑥
9		nfcv 2922	. . . . . . 7 ⊢ Ⅎ𝑓𝐺
10		bnj1466.11	. . . . . . . 8 ⊢ 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
11		nfcv 2922	. . . . . . . . . 10 ⊢ Ⅎ𝑓 pred(𝑥, 𝐴, 𝑅)
12	7, 11	nfres 5971	. . . . . . . . 9 ⊢ Ⅎ𝑓(𝑃 ↾ pred(𝑥, 𝐴, 𝑅))
13	8, 12	nfop 4848	. . . . . . . 8 ⊢ Ⅎ𝑓⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
14	10, 13	nfcxfr 2920	. . . . . . 7 ⊢ Ⅎ𝑓𝑍
15	9, 14	nffv 6895	. . . . . 6 ⊢ Ⅎ𝑓(𝐺‘𝑍)
16	8, 15	nfop 4848	. . . . 5 ⊢ Ⅎ𝑓⟨𝑥, (𝐺‘𝑍)⟩
17	16	nfsn 4667	. . . 4 ⊢ Ⅎ𝑓{⟨𝑥, (𝐺‘𝑍)⟩}
18	7, 17	nfun 4169	. . 3 ⊢ Ⅎ𝑓(𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})
19	1, 18	nfcxfr 2920	. 2 ⊢ Ⅎ𝑓𝑄
20	19	nfcrii 2917	1 ⊢ (𝑤 ∈ 𝑄 → ∀𝑓 𝑤 ∈ 𝑄)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103  ∀wal 1567   = wceq 1569  ∃wex 1811   ∈ wcel 2144  {cab 2738   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  {crab 3412  [wsbc 3738   ⊆ wss 3895   ∪ cun 4149  ∅c0 4278  {csn 4583  ⟨cop 4589  ∪ cuni 4866   class class class wbr 5102  dom cdm 5650   ↾ cres 5652   Fn wfn 6533  ‘cfv 6538   predc-bnj14 35319   FrSe w-bnj15 35323   trClc-bnj18 35325

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-xp 5656  df-res 5662  df-iota 6494  df-fv 6546

This theorem is used by:  bnj1463  35685  bnj1491  35687