Theorem bnj1489 35068

Description: Technical lemma for bnj60 35074. 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
bnj1489.1	⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1489.2	⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1489.3	⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
bnj1489.4	⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
bnj1489.5	⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}
bnj1489.6	⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))
bnj1489.7	⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))
bnj1489.8	⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)
bnj1489.9	⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
bnj1489.10	⊢ 𝑃 = ∪ 𝐻
bnj1489.11	⊢ 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1489.12	⊢ 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})

Assertion

Ref	Expression
bnj1489	⊢ (𝜒 → 𝑄 ∈ V)

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

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

Proof of Theorem bnj1489

Step	Hyp	Ref	Expression
1		bnj1489.12	. 2 ⊢ 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})
2		bnj1489.10	. . . 4 ⊢ 𝑃 = ∪ 𝐻
3		bnj1489.7	. . . . . . . 8 ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))
4		bnj1489.6	. . . . . . . . 9 ⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))
5		bnj1364 35040	. . . . . . . . . 10 ⊢ (𝑅 FrSe 𝐴 → 𝑅 Se 𝐴)
6		df-bnj13 34703	. . . . . . . . . 10 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 pred(𝑥, 𝐴, 𝑅) ∈ V)
7	5, 6	sylib 218	. . . . . . . . 9 ⊢ (𝑅 FrSe 𝐴 → ∀𝑥 ∈ 𝐴 pred(𝑥, 𝐴, 𝑅) ∈ V)
8	4, 7	bnj832 34770	. . . . . . . 8 ⊢ (𝜓 → ∀𝑥 ∈ 𝐴 pred(𝑥, 𝐴, 𝑅) ∈ V)
9	3, 8	bnj835 34771	. . . . . . 7 ⊢ (𝜒 → ∀𝑥 ∈ 𝐴 pred(𝑥, 𝐴, 𝑅) ∈ V)
10		bnj1489.5	. . . . . . . 8 ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}
11	10, 3	bnj1212 34811	. . . . . . 7 ⊢ (𝜒 → 𝑥 ∈ 𝐴)
12	9, 11	bnj1294 34829	. . . . . 6 ⊢ (𝜒 → pred(𝑥, 𝐴, 𝑅) ∈ V)
13		nfv 1915	. . . . . . . . 9 ⊢ Ⅎ𝑦𝜓
14		nfv 1915	. . . . . . . . 9 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐷
15		nfra1 3256	. . . . . . . . 9 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥
16	13, 14, 15	nf3an 1902	. . . . . . . 8 ⊢ Ⅎ𝑦(𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥)
17	3, 16	nfxfr 1854	. . . . . . 7 ⊢ Ⅎ𝑦𝜒
18	4	simplbi 497	. . . . . . . . . . 11 ⊢ (𝜓 → 𝑅 FrSe 𝐴)
19	3, 18	bnj835 34771	. . . . . . . . . 10 ⊢ (𝜒 → 𝑅 FrSe 𝐴)
20	19	adantr 480	. . . . . . . . 9 ⊢ ((𝜒 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → 𝑅 FrSe 𝐴)
21		bnj1489.1	. . . . . . . . . . 11 ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
22		bnj1489.2	. . . . . . . . . . 11 ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
23		bnj1489.3	. . . . . . . . . . 11 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
24		bnj1489.4	. . . . . . . . . . 11 ⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
25		bnj1489.8	. . . . . . . . . . 11 ⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)
26	21, 22, 23, 24, 10, 4, 3, 25	bnj1388 35045	. . . . . . . . . 10 ⊢ (𝜒 → ∀𝑦 ∈ pred (𝑥, 𝐴, 𝑅)∃𝑓𝜏′)
27	26	r19.21bi 3224	. . . . . . . . 9 ⊢ ((𝜒 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → ∃𝑓𝜏′)
28		nfv 1915	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥 𝑅 FrSe 𝐴
29		nfsbc1v 3756	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜏
30	25, 29	nfxfr 1854	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥𝜏′
31	30	nfex 2325	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥∃𝑓𝜏′
32	28, 31	nfan 1900	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑅 FrSe 𝐴 ∧ ∃𝑓𝜏′)
33	30	nfeuw 2588	. . . . . . . . . . 11 ⊢ Ⅎ𝑥∃!𝑓𝜏′
34	32, 33	nfim 1897	. . . . . . . . . 10 ⊢ Ⅎ𝑥((𝑅 FrSe 𝐴 ∧ ∃𝑓𝜏′) → ∃!𝑓𝜏′)
35		sneq 4583	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
36		bnj1318 35037	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → trCl(𝑥, 𝐴, 𝑅) = trCl(𝑦, 𝐴, 𝑅))
37	35, 36	uneq12d 4116	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
38	37	eqeq2d 2742	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → (dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) ↔ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
39	38	anbi2d 630	. . . . . . . . . . . . . 14 ⊢ (𝑥 = 𝑦 → ((𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))) ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))))
40	21, 22, 23, 24, 25	bnj1373 35042	. . . . . . . . . . . . . 14 ⊢ (𝜏′ ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
41	39, 40	bitr4di 289	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → ((𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))) ↔ 𝜏′))
42	41	exbidv 1922	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (∃𝑓(𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))) ↔ ∃𝑓𝜏′))
43	42	anbi2d 630	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ((𝑅 FrSe 𝐴 ∧ ∃𝑓(𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))) ↔ (𝑅 FrSe 𝐴 ∧ ∃𝑓𝜏′)))
44	41	eubidv 2581	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (∃!𝑓(𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))) ↔ ∃!𝑓𝜏′))
45	43, 44	imbi12d 344	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (((𝑅 FrSe 𝐴 ∧ ∃𝑓(𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))) → ∃!𝑓(𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))) ↔ ((𝑅 FrSe 𝐴 ∧ ∃𝑓𝜏′) → ∃!𝑓𝜏′)))
46		biid 261	. . . . . . . . . . 11 ⊢ ((𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))) ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
47	21, 22, 23, 46	bnj1321 35039	. . . . . . . . . 10 ⊢ ((𝑅 FrSe 𝐴 ∧ ∃𝑓(𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))) → ∃!𝑓(𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
48	34, 45, 47	chvarfv 2243	. . . . . . . . 9 ⊢ ((𝑅 FrSe 𝐴 ∧ ∃𝑓𝜏′) → ∃!𝑓𝜏′)
49	20, 27, 48	syl2anc 584	. . . . . . . 8 ⊢ ((𝜒 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → ∃!𝑓𝜏′)
50	49	ex 412	. . . . . . 7 ⊢ (𝜒 → (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) → ∃!𝑓𝜏′))
51	17, 50	ralrimi 3230	. . . . . 6 ⊢ (𝜒 → ∀𝑦 ∈ pred (𝑥, 𝐴, 𝑅)∃!𝑓𝜏′)
52		bnj1489.9	. . . . . . 7 ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
53	52	a1i 11	. . . . . 6 ⊢ (𝜒 → 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′})
54		biid 261	. . . . . . 7 ⊢ (( pred(𝑥, 𝐴, 𝑅) ∈ V ∧ ∀𝑦 ∈ pred (𝑥, 𝐴, 𝑅)∃!𝑓𝜏′ ∧ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}) ↔ ( pred(𝑥, 𝐴, 𝑅) ∈ V ∧ ∀𝑦 ∈ pred (𝑥, 𝐴, 𝑅)∃!𝑓𝜏′ ∧ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}))
55	54	bnj1366 34841	. . . . . 6 ⊢ (( pred(𝑥, 𝐴, 𝑅) ∈ V ∧ ∀𝑦 ∈ pred (𝑥, 𝐴, 𝑅)∃!𝑓𝜏′ ∧ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}) → 𝐻 ∈ V)
56	12, 51, 53, 55	syl3anc 1373	. . . . 5 ⊢ (𝜒 → 𝐻 ∈ V)
57	56	uniexd 7675	. . . 4 ⊢ (𝜒 → ∪ 𝐻 ∈ V)
58	2, 57	eqeltrid 2835	. . 3 ⊢ (𝜒 → 𝑃 ∈ V)
59		snex 5372	. . . 4 ⊢ {⟨𝑥, (𝐺‘𝑍)⟩} ∈ V
60	59	a1i 11	. . 3 ⊢ (𝜒 → {⟨𝑥, (𝐺‘𝑍)⟩} ∈ V)
61	58, 60	bnj1149 34804	. 2 ⊢ (𝜒 → (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩}) ∈ V)
62	1, 61	eqeltrid 2835	1 ⊢ (𝜒 → 𝑄 ∈ V)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1541  ∃wex 1780   ∈ wcel 2111  ∃!weu 2563  {cab 2709   ≠ wne 2928  ∀wral 3047  ∃wrex 3056  {crab 3395  Vcvv 3436  [wsbc 3736   ∪ cun 3895   ⊆ wss 3897  ∅c0 4280  {csn 4573  ⟨cop 4579  ∪ cuni 4856   class class class wbr 5089  dom cdm 5614   ↾ cres 5616   Fn wfn 6476  ‘cfv 6481   predc-bnj14 34700   Se w-bnj13 34702   FrSe w-bnj15 34704   trClc-bnj18 34706

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-reg 9478  ax-inf2 9531

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-ord 6309  df-on 6310  df-lim 6311  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-om 7797  df-1o 8385  df-bnj17 34699  df-bnj14 34701  df-bnj13 34703  df-bnj15 34705  df-bnj18 34707  df-bnj19 34709

This theorem is referenced by:  bnj1312  35070