Theorem bnj1489 35219

Description: Technical lemma for bnj60 35225. 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 35191	. . . . . . . . . 10 ⊢ (𝑅 FrSe 𝐴 → 𝑅 Se 𝐴)
6		df-bnj13 34855	. . . . . . . . . 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 34922	. . . . . . . 8 ⊢ (𝜓 → ∀𝑥 ∈ 𝐴 pred(𝑥, 𝐴, 𝑅) ∈ V)
9	3, 8	bnj835 34923	. . . . . . 7 ⊢ (𝜒 → ∀𝑥 ∈ 𝐴 pred(𝑥, 𝐴, 𝑅) ∈ V)
10		bnj1489.5	. . . . . . . 8 ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}
11	10, 3	bnj1212 34962	. . . . . . 7 ⊢ (𝜒 → 𝑥 ∈ 𝐴)
12	9, 11	bnj1294 34980	. . . . . 6 ⊢ (𝜒 → pred(𝑥, 𝐴, 𝑅) ∈ V)
13		nfv 1916	. . . . . . . . 9 ⊢ Ⅎ𝑦𝜓
14		nfv 1916	. . . . . . . . 9 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐷
15		nfra1 3262	. . . . . . . . 9 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥
16	13, 14, 15	nf3an 1903	. . . . . . . 8 ⊢ Ⅎ𝑦(𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥)
17	3, 16	nfxfr 1855	. . . . . . 7 ⊢ Ⅎ𝑦𝜒
18	4	simplbi 496	. . . . . . . . . . 11 ⊢ (𝜓 → 𝑅 FrSe 𝐴)
19	3, 18	bnj835 34923	. . . . . . . . . 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 35196	. . . . . . . . . 10 ⊢ (𝜒 → ∀𝑦 ∈ pred (𝑥, 𝐴, 𝑅)∃𝑓𝜏′)
27	26	r19.21bi 3230	. . . . . . . . 9 ⊢ ((𝜒 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → ∃𝑓𝜏′)
28		nfv 1916	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥 𝑅 FrSe 𝐴
29		nfsbc1v 3749	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜏
30	25, 29	nfxfr 1855	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥𝜏′
31	30	nfex 2330	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥∃𝑓𝜏′
32	28, 31	nfan 1901	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑅 FrSe 𝐴 ∧ ∃𝑓𝜏′)
33	30	nfeuw 2594	. . . . . . . . . . 11 ⊢ Ⅎ𝑥∃!𝑓𝜏′
34	32, 33	nfim 1898	. . . . . . . . . 10 ⊢ Ⅎ𝑥((𝑅 FrSe 𝐴 ∧ ∃𝑓𝜏′) → ∃!𝑓𝜏′)
35		sneq 4578	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
36		bnj1318 35188	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → trCl(𝑥, 𝐴, 𝑅) = trCl(𝑦, 𝐴, 𝑅))
37	35, 36	uneq12d 4110	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
38	37	eqeq2d 2748	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → (dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) ↔ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
39	38	anbi2d 631	. . . . . . . . . . . . . 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 35193	. . . . . . . . . . . . . 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 1923	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (∃𝑓(𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))) ↔ ∃𝑓𝜏′))
43	42	anbi2d 631	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ((𝑅 FrSe 𝐴 ∧ ∃𝑓(𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))) ↔ (𝑅 FrSe 𝐴 ∧ ∃𝑓𝜏′)))
44	41	eubidv 2587	. . . . . . . . . . 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 35190	. . . . . . . . . 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 2248	. . . . . . . . 9 ⊢ ((𝑅 FrSe 𝐴 ∧ ∃𝑓𝜏′) → ∃!𝑓𝜏′)
49	20, 27, 48	syl2anc 585	. . . . . . . 8 ⊢ ((𝜒 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → ∃!𝑓𝜏′)
50	49	ex 412	. . . . . . 7 ⊢ (𝜒 → (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) → ∃!𝑓𝜏′))
51	17, 50	ralrimi 3236	. . . . . 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 34992	. . . . . 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 1374	. . . . 5 ⊢ (𝜒 → 𝐻 ∈ V)
57	56	uniexd 7687	. . . 4 ⊢ (𝜒 → ∪ 𝐻 ∈ V)
58	2, 57	eqeltrid 2841	. . 3 ⊢ (𝜒 → 𝑃 ∈ V)
59		snex 5374	. . . 4 ⊢ {⟨𝑥, (𝐺‘𝑍)⟩} ∈ V
60	59	a1i 11	. . 3 ⊢ (𝜒 → {⟨𝑥, (𝐺‘𝑍)⟩} ∈ V)
61	58, 60	bnj1149 34955	. 2 ⊢ (𝜒 → (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩}) ∈ V)
62	1, 61	eqeltrid 2841	1 ⊢ (𝜒 → 𝑄 ∈ V)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1087   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∃!weu 2569  {cab 2715   ≠ wne 2933  ∀wral 3052  ∃wrex 3062  {crab 3390  Vcvv 3430  [wsbc 3729   ∪ cun 3888   ⊆ wss 3890  ∅c0 4274  {csn 4568  ⟨cop 4574  ∪ cuni 4851   class class class wbr 5086  dom cdm 5622   ↾ cres 5624   Fn wfn 6485  ‘cfv 6490   predc-bnj14 34852   Se w-bnj13 34854   FrSe w-bnj15 34856   trClc-bnj18 34858

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5300  ax-pr 5368  ax-un 7680  ax-reg 9498  ax-inf2 9551

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-pss 3910  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-tr 5194  df-id 5517  df-eprel 5522  df-po 5530  df-so 5531  df-fr 5575  df-we 5577  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-ord 6318  df-on 6319  df-lim 6320  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-om 7809  df-1o 8396  df-bnj17 34851  df-bnj14 34853  df-bnj13 34855  df-bnj15 34857  df-bnj18 34859  df-bnj19 34861

This theorem is referenced by:  bnj1312  35221