Theorem bnj1388 32913

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
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 1918	. . . 4 ⊢ Ⅎ𝑦𝜓
3		nfv 1918	. . . 4 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐷
4		nfra1 3142	. . . 4 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥
5	2, 3, 4	nf3an 1905	. . 3 ⊢ Ⅎ𝑦(𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥)
6	1, 5	nfxfr 1856	. 2 ⊢ Ⅎ𝑦𝜒
7		bnj1152 32878	. . . . . 6 ⊢ (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ↔ (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥))
8	7	simplbi 497	. . . . 5 ⊢ (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) → 𝑦 ∈ 𝐴)
9	8	adantl 481	. . . 4 ⊢ ((𝜒 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → 𝑦 ∈ 𝐴)
10	7	biimpi 215	. . . . . . . . 9 ⊢ (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) → (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥))
11	10	adantl 481	. . . . . . . 8 ⊢ ((𝜒 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → (𝑦 ∈ 𝐴 ∧ 𝑦𝑅𝑥))
12	11	simprd 495	. . . . . . 7 ⊢ ((𝜒 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → 𝑦𝑅𝑥)
13	1	simp3bi 1145	. . . . . . . 8 ⊢ (𝜒 → ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥)
14	13	adantr 480	. . . . . . 7 ⊢ ((𝜒 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥)
15		df-ral 3068	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥 ↔ ∀𝑦(𝑦 ∈ 𝐷 → ¬ 𝑦𝑅𝑥))
16		con2b 359	. . . . . . . . . 10 ⊢ ((𝑦 ∈ 𝐷 → ¬ 𝑦𝑅𝑥) ↔ (𝑦𝑅𝑥 → ¬ 𝑦 ∈ 𝐷))
17	16	albii 1823	. . . . . . . . 9 ⊢ (∀𝑦(𝑦 ∈ 𝐷 → ¬ 𝑦𝑅𝑥) ↔ ∀𝑦(𝑦𝑅𝑥 → ¬ 𝑦 ∈ 𝐷))
18	15, 17	bitri 274	. . . . . . . 8 ⊢ (∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥 ↔ ∀𝑦(𝑦𝑅𝑥 → ¬ 𝑦 ∈ 𝐷))
19		sp 2178	. . . . . . . . 9 ⊢ (∀𝑦(𝑦𝑅𝑥 → ¬ 𝑦 ∈ 𝐷) → (𝑦𝑅𝑥 → ¬ 𝑦 ∈ 𝐷))
20	19	impcom 407	. . . . . . . 8 ⊢ ((𝑦𝑅𝑥 ∧ ∀𝑦(𝑦𝑅𝑥 → ¬ 𝑦 ∈ 𝐷)) → ¬ 𝑦 ∈ 𝐷)
21	18, 20	sylan2b 593	. . . . . . 7 ⊢ ((𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥) → ¬ 𝑦 ∈ 𝐷)
22	12, 14, 21	syl2anc 583	. . . . . 6 ⊢ ((𝜒 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → ¬ 𝑦 ∈ 𝐷)
23		bnj1388.5	. . . . . . . 8 ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}
24	23	eleq2i 2830	. . . . . . 7 ⊢ (𝑦 ∈ 𝐷 ↔ 𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏})
25		nfcv 2906	. . . . . . . 8 ⊢ Ⅎ𝑥𝑦
26		nfcv 2906	. . . . . . . 8 ⊢ Ⅎ𝑥𝐴
27		bnj1388.8	. . . . . . . . . . 11 ⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)
28		nfsbc1v 3731	. . . . . . . . . . 11 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜏
29	27, 28	nfxfr 1856	. . . . . . . . . 10 ⊢ Ⅎ𝑥𝜏′
30	29	nfex 2322	. . . . . . . . 9 ⊢ Ⅎ𝑥∃𝑓𝜏′
31	30	nfn 1861	. . . . . . . 8 ⊢ Ⅎ𝑥 ¬ ∃𝑓𝜏′
32		sbceq1a 3722	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (𝜏 ↔ [𝑦 / 𝑥]𝜏))
33	32, 27	bitr4di 288	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝜏 ↔ 𝜏′))
34	33	exbidv 1925	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (∃𝑓𝜏 ↔ ∃𝑓𝜏′))
35	34	notbid 317	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (¬ ∃𝑓𝜏 ↔ ¬ ∃𝑓𝜏′))
36	25, 26, 31, 35	elrabf 3613	. . . . . . 7 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏} ↔ (𝑦 ∈ 𝐴 ∧ ¬ ∃𝑓𝜏′))
37	24, 36	bitri 274	. . . . . 6 ⊢ (𝑦 ∈ 𝐷 ↔ (𝑦 ∈ 𝐴 ∧ ¬ ∃𝑓𝜏′))
38	22, 37	sylnib 327	. . . . 5 ⊢ ((𝜒 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → ¬ (𝑦 ∈ 𝐴 ∧ ¬ ∃𝑓𝜏′))
39		iman 401	. . . . 5 ⊢ ((𝑦 ∈ 𝐴 → ∃𝑓𝜏′) ↔ ¬ (𝑦 ∈ 𝐴 ∧ ¬ ∃𝑓𝜏′))
40	38, 39	sylibr 233	. . . 4 ⊢ ((𝜒 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → (𝑦 ∈ 𝐴 → ∃𝑓𝜏′))
41	9, 40	mpd 15	. . 3 ⊢ ((𝜒 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → ∃𝑓𝜏′)
42	41	ex 412	. 2 ⊢ (𝜒 → (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) → ∃𝑓𝜏′))
43	6, 42	ralrimi 3139	1 ⊢ (𝜒 → ∀𝑦 ∈ pred (𝑥, 𝐴, 𝑅)∃𝑓𝜏′)

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   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-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-br 5071  df-bnj14 32568

This theorem is referenced by:  bnj1398  32914  bnj1489  32936