Theorem bnj1466 31450

Description: Technical lemma for bnj60 31459. 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 31221	. . . . . . 7 ⊢ (𝑤 ∈ 𝐻 → ∀𝑓 𝑤 ∈ 𝐻)
5	4	nfcii 2894	. . . . . 6 ⊢ Ⅎ𝑓𝐻
6	5	nfuni 4595	. . . . 5 ⊢ Ⅎ𝑓∪ 𝐻
7	2, 6	nfcxfr 2901	. . . 4 ⊢ Ⅎ𝑓𝑃
8		nfcv 2903	. . . . . 6 ⊢ Ⅎ𝑓𝑥
9		nfcv 2903	. . . . . . 7 ⊢ Ⅎ𝑓𝐺
10		bnj1466.11	. . . . . . . 8 ⊢ 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
11		nfcv 2903	. . . . . . . . . 10 ⊢ Ⅎ𝑓 pred(𝑥, 𝐴, 𝑅)
12	7, 11	nfres 5554	. . . . . . . . 9 ⊢ Ⅎ𝑓(𝑃 ↾ pred(𝑥, 𝐴, 𝑅))
13	8, 12	nfop 4570	. . . . . . . 8 ⊢ Ⅎ𝑓⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
14	10, 13	nfcxfr 2901	. . . . . . 7 ⊢ Ⅎ𝑓𝑍
15	9, 14	nffv 6361	. . . . . 6 ⊢ Ⅎ𝑓(𝐺‘𝑍)
16	8, 15	nfop 4570	. . . . 5 ⊢ Ⅎ𝑓⟨𝑥, (𝐺‘𝑍)⟩
17	16	nfsn 4387	. . . 4 ⊢ Ⅎ𝑓{⟨𝑥, (𝐺‘𝑍)⟩}
18	7, 17	nfun 3913	. . 3 ⊢ Ⅎ𝑓(𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})
19	1, 18	nfcxfr 2901	. 2 ⊢ Ⅎ𝑓𝑄
20	19	nfcrii 2896	1 ⊢ (𝑤 ∈ 𝑄 → ∀𝑓 𝑤 ∈ 𝑄)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1072  ∀wal 1630   = wceq 1632  ∃wex 1853   ∈ wcel 2140  {cab 2747   ≠ wne 2933  ∀wral 3051  ∃wrex 3052  {crab 3055  [wsbc 3577   ∪ cun 3714   ⊆ wss 3716  ∅c0 4059  {csn 4322  ⟨cop 4328  ∪ cuni 4589   class class class wbr 4805  dom cdm 5267   ↾ cres 5269   Fn wfn 6045  ‘cfv 6050   predc-bnj14 31085   FrSe w-bnj15 31089   trClc-bnj18 31091

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-nul 4060  df-if 4232  df-sn 4323  df-pr 4325  df-op 4329  df-uni 4590  df-br 4806  df-opab 4866  df-xp 5273  df-res 5279  df-iota 6013  df-fv 6058

This theorem is referenced by:  bnj1463  31452  bnj1491  31454