Theorem bnj1489 32330

Description: Technical lemma for bnj60 32336. 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 32302	. . . . . . . . . 10 ⊢ (𝑅 FrSe 𝐴 → 𝑅 Se 𝐴)
6		df-bnj13 31963	. . . . . . . . . 10 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 pred(𝑥, 𝐴, 𝑅) ∈ V)
7	5, 6	sylib 220	. . . . . . . . 9 ⊢ (𝑅 FrSe 𝐴 → ∀𝑥 ∈ 𝐴 pred(𝑥, 𝐴, 𝑅) ∈ V)
8	4, 7	bnj832 32031	. . . . . . . 8 ⊢ (𝜓 → ∀𝑥 ∈ 𝐴 pred(𝑥, 𝐴, 𝑅) ∈ V)
9	3, 8	bnj835 32032	. . . . . . 7 ⊢ (𝜒 → ∀𝑥 ∈ 𝐴 pred(𝑥, 𝐴, 𝑅) ∈ V)
10		bnj1489.5	. . . . . . . 8 ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}
11	10, 3	bnj1212 32073	. . . . . . 7 ⊢ (𝜒 → 𝑥 ∈ 𝐴)
12	9, 11	bnj1294 32091	. . . . . 6 ⊢ (𝜒 → pred(𝑥, 𝐴, 𝑅) ∈ V)
13		nfv 1915	. . . . . . . . 9 ⊢ Ⅎ𝑦𝜓
14		nfv 1915	. . . . . . . . 9 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐷
15		nfra1 3221	. . . . . . . . 9 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥
16	13, 14, 15	nf3an 1902	. . . . . . . 8 ⊢ Ⅎ𝑦(𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥)
17	3, 16	nfxfr 1853	. . . . . . 7 ⊢ Ⅎ𝑦𝜒
18	4	simplbi 500	. . . . . . . . . . 11 ⊢ (𝜓 → 𝑅 FrSe 𝐴)
19	3, 18	bnj835 32032	. . . . . . . . . 10 ⊢ (𝜒 → 𝑅 FrSe 𝐴)
20	19	adantr 483	. . . . . . . . 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 32307	. . . . . . . . . 10 ⊢ (𝜒 → ∀𝑦 ∈ pred (𝑥, 𝐴, 𝑅)∃𝑓𝜏′)
27	26	r19.21bi 3210	. . . . . . . . 9 ⊢ ((𝜒 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → ∃𝑓𝜏′)
28		nfv 1915	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥 𝑅 FrSe 𝐴
29		nfsbc1v 3794	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜏
30	25, 29	nfxfr 1853	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥𝜏′
31	30	nfex 2343	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥∃𝑓𝜏′
32	28, 31	nfan 1900	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑅 FrSe 𝐴 ∧ ∃𝑓𝜏′)
33	30	nfeuw 2679	. . . . . . . . . . 11 ⊢ Ⅎ𝑥∃!𝑓𝜏′
34	32, 33	nfim 1897	. . . . . . . . . 10 ⊢ Ⅎ𝑥((𝑅 FrSe 𝐴 ∧ ∃𝑓𝜏′) → ∃!𝑓𝜏′)
35		sneq 4579	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
36		bnj1318 32299	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → trCl(𝑥, 𝐴, 𝑅) = trCl(𝑦, 𝐴, 𝑅))
37	35, 36	uneq12d 4142	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
38	37	eqeq2d 2834	. . . . . . . . . . . . . . 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 32304	. . . . . . . . . . . . . 14 ⊢ (𝜏′ ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
41	39, 40	syl6bbr 291	. . . . . . . . . . . . 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 2672	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (∃!𝑓(𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))) ↔ ∃!𝑓𝜏′))
45	43, 44	imbi12d 347	. . . . . . . . . 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 263	. . . . . . . . . . 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 32301	. . . . . . . . . 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 2242	. . . . . . . . 9 ⊢ ((𝑅 FrSe 𝐴 ∧ ∃𝑓𝜏′) → ∃!𝑓𝜏′)
49	20, 27, 48	syl2anc 586	. . . . . . . 8 ⊢ ((𝜒 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → ∃!𝑓𝜏′)
50	49	ex 415	. . . . . . 7 ⊢ (𝜒 → (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) → ∃!𝑓𝜏′))
51	17, 50	ralrimi 3218	. . . . . 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 263	. . . . . . 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 32103	. . . . . 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 1367	. . . . 5 ⊢ (𝜒 → 𝐻 ∈ V)
57	56	uniexd 7470	. . . 4 ⊢ (𝜒 → ∪ 𝐻 ∈ V)
58	2, 57	eqeltrid 2919	. . 3 ⊢ (𝜒 → 𝑃 ∈ V)
59		snex 5334	. . . 4 ⊢ {⟨𝑥, (𝐺‘𝑍)⟩} ∈ V
60	59	a1i 11	. . 3 ⊢ (𝜒 → {⟨𝑥, (𝐺‘𝑍)⟩} ∈ V)
61	58, 60	bnj1149 32066	. 2 ⊢ (𝜒 → (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩}) ∈ V)
62	1, 61	eqeltrid 2919	1 ⊢ (𝜒 → 𝑄 ∈ V)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1537  ∃wex 1780   ∈ wcel 2114  ∃!weu 2653  {cab 2801   ≠ wne 3018  ∀wral 3140  ∃wrex 3141  {crab 3144  Vcvv 3496  [wsbc 3774   ∪ cun 3936   ⊆ wss 3938  ∅c0 4293  {csn 4569  ⟨cop 4575  ∪ cuni 4840   class class class wbr 5068  dom cdm 5557   ↾ cres 5559   Fn wfn 6352  ‘cfv 6357   predc-bnj14 31960   Se w-bnj13 31962   FrSe w-bnj15 31964   trClc-bnj18 31966

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-rep 5192  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463  ax-reg 9058  ax-inf2 9106

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-fal 1550  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-reu 3147  df-rab 3149  df-v 3498  df-sbc 3775  df-csb 3886  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-iun 4923  df-br 5069  df-opab 5131  df-mpt 5149  df-tr 5175  df-id 5462  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-fv 6365  df-om 7583  df-1o 8104  df-bnj17 31959  df-bnj14 31961  df-bnj13 31963  df-bnj15 31965  df-bnj18 31967  df-bnj19 31969

This theorem is referenced by:  bnj1312  32332