Theorem bnj1388 35529

Description: Technical lemma for bnj60 35558. 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
bnj1388.1	⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1388.2	⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1388.3	⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
bnj1388.4	⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
bnj1388.5	⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}
bnj1388.6	⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))
bnj1388.7	⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))
bnj1388.8	⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)

Assertion

Ref	Expression
bnj1388	⊢ (𝜒 → ∀𝑦 ∈ pred (𝑥, 𝐴, 𝑅)∃𝑓𝜏′)

Distinct variable groups:   𝑥,𝐴,𝑦   𝐵,𝑓   𝑦,𝐷   𝑥,𝑅,𝑦   𝑓,𝑑,𝑥   𝑦,𝑓   𝜓,𝑦   𝜏,𝑦

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

Proof of Theorem bnj1388

Step	Hyp	Ref	Expression
1		bnj1388.7	. . 3 ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))
2		nfv 1947	. . . 4 ⊢ Ⅎ𝑦𝜓
3		nfv 1947	. . . 4 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐷
4		nfra1 3288	. . . 4 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥
5	2, 3, 4	nf3an 1934	. . 3 ⊢ Ⅎ𝑦(𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥)
6	1, 5	nfxfr 1886	. 2 ⊢ Ⅎ𝑦𝜒
7		bnj1152 35494	. . . . . 6 ⊢ (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ↔ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥))
8	7	simplbi 502	. . . . 5 ⊢ (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) → 𝑦 ∈ 𝐴)
9	8	adantl 487	. . . 4 ⊢ ((𝜒 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → 𝑦 ∈ 𝐴)
10	7	bilani 510	. . . . . . . 8 ⊢ ((𝜒 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥))
11	10	simprd 501	. . . . . . 7 ⊢ ((𝜒 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → 𝑦𝑅𝑥)
12	1	simp3bi 1165	. . . . . . . 8 ⊢ (𝜒 → ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥)
13	12	adantr 486	. . . . . . 7 ⊢ ((𝜒 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥)
14		df-ral 3079	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥 ↔ ∀𝑦(𝑦 ∈ 𝐷 → ¬ 𝑦𝑅𝑥))
15		con2b 362	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐷 → ¬ 𝑦𝑅𝑥) ↔ (𝑦𝑅𝑥 → ¬ 𝑦 ∈ 𝐷))
16	15	albii 1852	. . . . . . . . 9 ⊢ (∀𝑦(𝑦 ∈ 𝐷 → ¬ 𝑦𝑅𝑥) ↔ ∀𝑦(𝑦𝑅𝑥 → ¬ 𝑦 ∈ 𝐷))
17	14, 16	bitri 278	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥 ↔ ∀𝑦(𝑦𝑅𝑥 → ¬ 𝑦 ∈ 𝐷))
18		sp 2221	. . . . . . . . 9 ⊢ (∀𝑦(𝑦𝑅𝑥 → ¬ 𝑦 ∈ 𝐷) → (𝑦𝑅𝑥 → ¬ 𝑦 ∈ 𝐷))
19	18	impcom 413	. . . . . . . 8 ⊢ ((𝑦𝑅𝑥 ∧ ∀𝑦(𝑦𝑅𝑥 → ¬ 𝑦 ∈ 𝐷)) → ¬ 𝑦 ∈ 𝐷)
20	17, 19	sylan2b 606	. . . . . . 7 ⊢ ((𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥) → ¬ 𝑦 ∈ 𝐷)
21	11, 13, 20	syl2anc 596	. . . . . 6 ⊢ ((𝜒 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → ¬ 𝑦 ∈ 𝐷)
22		bnj1388.5	. . . . . . . 8 ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}
23	22	eleq2i 2854	. . . . . . 7 ⊢ (𝑦 ∈ 𝐷 ↔ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏})
24		nfcv 2924	. . . . . . . 8 ⊢ Ⅎ𝑥𝑦
25		nfcv 2924	. . . . . . . 8 ⊢ Ⅎ𝑥𝐴
26		bnj1388.8	. . . . . . . . . . 11 ⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)
27		nfsbc1v 3762	. . . . . . . . . . 11 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜏
28	26, 27	nfxfr 1886	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝜏′
29	28	nfex 2356	. . . . . . . . 9 ⊢ Ⅎ𝑥∃𝑓𝜏′
30	29	nfn 1890	. . . . . . . 8 ⊢ Ⅎ𝑥 ¬ ∃𝑓𝜏′
31		sbceq1a 3753	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝜏 ↔ [𝑦 / 𝑥]𝜏))
32	31, 26	bitr4di 292	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝜏 ↔ 𝜏′))
33	32	exbidv 1954	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (∃𝑓𝜏 ↔ ∃𝑓𝜏′))
34	33	notbid 321	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (¬ ∃𝑓𝜏 ↔ ¬ ∃𝑓𝜏′))
35	24, 25, 30, 34	elrabf 3645	. . . . . . 7 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏} ↔ (𝑦 ∈ 𝐴 ∧ ¬ ∃𝑓𝜏′))
36	23, 35	bitri 278	. . . . . 6 ⊢ (𝑦 ∈ 𝐷 ↔ (𝑦 ∈ 𝐴 ∧ ¬ ∃𝑓𝜏′))
37	21, 36	sylnib 331	. . . . 5 ⊢ ((𝜒 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → ¬ (𝑦 ∈ 𝐴 ∧ ¬ ∃𝑓𝜏′))
38		iman 407	. . . . 5 ⊢ ((𝑦 ∈ 𝐴 → ∃𝑓𝜏′) ↔ ¬ (𝑦 ∈ 𝐴 ∧ ¬ ∃𝑓𝜏′))
39	37, 38	sylibr 237	. . . 4 ⊢ ((𝜒 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → (𝑦 ∈ 𝐴 → ∃𝑓𝜏′))
40	9, 39	mpd 16	. . 3 ⊢ ((𝜒 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → ∃𝑓𝜏′)
41	40	ex 418	. 2 ⊢ (𝜒 → (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) → ∃𝑓𝜏′))
42	6, 41	ralrimi 3262	1 ⊢ (𝜒 → ∀𝑦 ∈ pred (𝑥, 𝐴, 𝑅)∃𝑓𝜏′)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2145  {cab 2740   ≠ wne 2957  ∀wral 3078  ∃wrex 3088  {crab 3414  [wsbc 3742   ∪ cun 3900   ⊆ wss 3902  ∅c0 4282  {csn 4587  ⟨cop 4593   class class class wbr 5107  dom cdm 5659   ↾ cres 5661   Fn wfn 6532  ‘cfv 6537   predc-bnj14 35185   FrSe w-bnj15 35189   trClc-bnj18 35191

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ral 3079  df-rab 3415  df-v 3455  df-sbc 3743  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-bnj14 35186

This theorem is used by:  bnj1398  35530  bnj1489  35552