Theorem bnj1489 31124

Description: Technical lemma for bnj60 31130. 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 31096	. . . . . . . . . 10 ⊢ (𝑅 FrSe 𝐴 → 𝑅 Se 𝐴)
6		df-bnj13 30757	. . . . . . . . . 10 ⊢ (𝑅 Se 𝐴 ↔ ∀𝑥 ∈ 𝐴 pred(𝑥, 𝐴, 𝑅) ∈ V)
7	5, 6	sylib 208	. . . . . . . . 9 ⊢ (𝑅 FrSe 𝐴 → ∀𝑥 ∈ 𝐴 pred(𝑥, 𝐴, 𝑅) ∈ V)
8	4, 7	bnj832 30828	. . . . . . . 8 ⊢ (𝜓 → ∀𝑥 ∈ 𝐴 pred(𝑥, 𝐴, 𝑅) ∈ V)
9	3, 8	bnj835 30829	. . . . . . 7 ⊢ (𝜒 → ∀𝑥 ∈ 𝐴 pred(𝑥, 𝐴, 𝑅) ∈ V)
10		bnj1489.5	. . . . . . . 8 ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}
11	10, 3	bnj1212 30870	. . . . . . 7 ⊢ (𝜒 → 𝑥 ∈ 𝐴)
12	9, 11	bnj1294 30888	. . . . . 6 ⊢ (𝜒 → pred(𝑥, 𝐴, 𝑅) ∈ V)
13		nfv 1843	. . . . . . . . 9 ⊢ Ⅎ𝑦𝜓
14		nfv 1843	. . . . . . . . 9 ⊢ Ⅎ𝑦 𝑥 ∈ 𝐷
15		nfra1 2941	. . . . . . . . 9 ⊢ Ⅎ𝑦∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥
16	13, 14, 15	nf3an 1831	. . . . . . . 8 ⊢ Ⅎ𝑦(𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥)
17	3, 16	nfxfr 1779	. . . . . . 7 ⊢ Ⅎ𝑦𝜒
18	4	simplbi 476	. . . . . . . . . . 11 ⊢ (𝜓 → 𝑅 FrSe 𝐴)
19	3, 18	bnj835 30829	. . . . . . . . . 10 ⊢ (𝜒 → 𝑅 FrSe 𝐴)
20	19	adantr 481	. . . . . . . . 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 31101	. . . . . . . . . 10 ⊢ (𝜒 → ∀𝑦 ∈ pred (𝑥, 𝐴, 𝑅)∃𝑓𝜏′)
27	26	r19.21bi 2932	. . . . . . . . 9 ⊢ ((𝜒 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → ∃𝑓𝜏′)
28		nfv 1843	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥 𝑅 FrSe 𝐴
29		nfsbc1v 3455	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑥[𝑦 / 𝑥]𝜏
30	25, 29	nfxfr 1779	. . . . . . . . . . . . 13 ⊢ Ⅎ𝑥𝜏′
31	30	nfex 2154	. . . . . . . . . . . 12 ⊢ Ⅎ𝑥∃𝑓𝜏′
32	28, 31	nfan 1828	. . . . . . . . . . 11 ⊢ Ⅎ𝑥(𝑅 FrSe 𝐴 ∧ ∃𝑓𝜏′)
33	30	nfeu 2486	. . . . . . . . . . 11 ⊢ Ⅎ𝑥∃!𝑓𝜏′
34	32, 33	nfim 1825	. . . . . . . . . 10 ⊢ Ⅎ𝑥((𝑅 FrSe 𝐴 ∧ ∃𝑓𝜏′) → ∃!𝑓𝜏′)
35		sneq 4187	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦})
36		bnj1318 31093	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = 𝑦 → trCl(𝑥, 𝐴, 𝑅) = trCl(𝑦, 𝐴, 𝑅))
37	35, 36	uneq12d 3768	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = 𝑦 → ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
38	37	eqeq2d 2632	. . . . . . . . . . . . . . 15 ⊢ (𝑥 = 𝑦 → (dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) ↔ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
39	38	anbi2d 740	. . . . . . . . . . . . . 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 31098	. . . . . . . . . . . . . 14 ⊢ (𝜏′ ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
41	39, 40	syl6bbr 278	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → ((𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))) ↔ 𝜏′))
42	41	exbidv 1850	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (∃𝑓(𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))) ↔ ∃𝑓𝜏′))
43	42	anbi2d 740	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → ((𝑅 FrSe 𝐴 ∧ ∃𝑓(𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))) ↔ (𝑅 FrSe 𝐴 ∧ ∃𝑓𝜏′)))
44	41	eubidv 2490	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑦 → (∃!𝑓(𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))) ↔ ∃!𝑓𝜏′))
45	43, 44	imbi12d 334	. . . . . . . . . 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 251	. . . . . . . . . . 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 31095	. . . . . . . . . 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	chvar 2262	. . . . . . . . 9 ⊢ ((𝑅 FrSe 𝐴 ∧ ∃𝑓𝜏′) → ∃!𝑓𝜏′)
49	20, 27, 48	syl2anc 693	. . . . . . . 8 ⊢ ((𝜒 ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → ∃!𝑓𝜏′)
50	49	ex 450	. . . . . . 7 ⊢ (𝜒 → (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) → ∃!𝑓𝜏′))
51	17, 50	ralrimi 2957	. . . . . 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 251	. . . . . . 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 30900	. . . . . 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 1326	. . . . 5 ⊢ (𝜒 → 𝐻 ∈ V)
57		uniexg 6955	. . . . 5 ⊢ (𝐻 ∈ V → ∪ 𝐻 ∈ V)
58	56, 57	syl 17	. . . 4 ⊢ (𝜒 → ∪ 𝐻 ∈ V)
59	2, 58	syl5eqel 2705	. . 3 ⊢ (𝜒 → 𝑃 ∈ V)
60		snex 4908	. . . 4 ⊢ {⟨𝑥, (𝐺‘𝑍)⟩} ∈ V
61	60	a1i 11	. . 3 ⊢ (𝜒 → {⟨𝑥, (𝐺‘𝑍)⟩} ∈ V)
62	59, 61	bnj1149 30863	. 2 ⊢ (𝜒 → (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩}) ∈ V)
63	1, 62	syl5eqel 2705	1 ⊢ (𝜒 → 𝑄 ∈ V)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ∃!weu 2470  {cab 2608   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  {crab 2916  Vcvv 3200  [wsbc 3435   ∪ cun 3572   ⊆ wss 3574  ∅c0 3915  {csn 4177  ⟨cop 4183  ∪ cuni 4436   class class class wbr 4653  dom cdm 5114   ↾ cres 5116   Fn wfn 5883  ‘cfv 5888   predc-bnj14 30754   Se w-bnj13 30756   FrSe w-bnj15 30758   trClc-bnj18 30760

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-reg 8497  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-fal 1489  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-om 7066  df-1o 7560  df-bnj17 30753  df-bnj14 30755  df-bnj13 30757  df-bnj15 30759  df-bnj18 30761  df-bnj19 30763

This theorem is referenced by:  bnj1312  31126