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Theorem bnj1489 31250
 Description: Technical lemma for bnj60 31256. 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 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1489.2 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1489.3 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
bnj1489.4 (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
bnj1489.5 𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}
bnj1489.6 (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))
bnj1489.7 (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 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, (𝐺𝑍)⟩})
Assertion
Ref Expression
bnj1489 (𝜒𝑄 ∈ 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
StepHypRef Expression
1 bnj1489.12 . 2 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})
2 bnj1489.10 . . . 4 𝑃 = 𝐻
3 bnj1489.7 . . . . . . . 8 (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))
4 bnj1489.6 . . . . . . . . 9 (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))
5 bnj1364 31222 . . . . . . . . . 10 (𝑅 FrSe 𝐴𝑅 Se 𝐴)
6 df-bnj13 30885 . . . . . . . . . 10 (𝑅 Se 𝐴 ↔ ∀𝑥𝐴 pred(𝑥, 𝐴, 𝑅) ∈ V)
75, 6sylib 208 . . . . . . . . 9 (𝑅 FrSe 𝐴 → ∀𝑥𝐴 pred(𝑥, 𝐴, 𝑅) ∈ V)
84, 7bnj832 30954 . . . . . . . 8 (𝜓 → ∀𝑥𝐴 pred(𝑥, 𝐴, 𝑅) ∈ V)
93, 8bnj835 30955 . . . . . . 7 (𝜒 → ∀𝑥𝐴 pred(𝑥, 𝐴, 𝑅) ∈ V)
10 bnj1489.5 . . . . . . . 8 𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}
1110, 3bnj1212 30996 . . . . . . 7 (𝜒𝑥𝐴)
129, 11bnj1294 31014 . . . . . 6 (𝜒 → pred(𝑥, 𝐴, 𝑅) ∈ V)
13 nfv 1883 . . . . . . . . 9 𝑦𝜓
14 nfv 1883 . . . . . . . . 9 𝑦 𝑥𝐷
15 nfra1 2970 . . . . . . . . 9 𝑦𝑦𝐷 ¬ 𝑦𝑅𝑥
1613, 14, 15nf3an 1871 . . . . . . . 8 𝑦(𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥)
173, 16nfxfr 1819 . . . . . . 7 𝑦𝜒
184simplbi 475 . . . . . . . . . . 11 (𝜓𝑅 FrSe 𝐴)
193, 18bnj835 30955 . . . . . . . . . 10 (𝜒𝑅 FrSe 𝐴)
2019adantr 480 . . . . . . . . 9 ((𝜒𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → 𝑅 FrSe 𝐴)
21 bnj1489.1 . . . . . . . . . . 11 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
22 bnj1489.2 . . . . . . . . . . 11 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
23 bnj1489.3 . . . . . . . . . . 11 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
24 bnj1489.4 . . . . . . . . . . 11 (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
25 bnj1489.8 . . . . . . . . . . 11 (𝜏′[𝑦 / 𝑥]𝜏)
2621, 22, 23, 24, 10, 4, 3, 25bnj1388 31227 . . . . . . . . . 10 (𝜒 → ∀𝑦 ∈ pred (𝑥, 𝐴, 𝑅)∃𝑓𝜏′)
2726r19.21bi 2961 . . . . . . . . 9 ((𝜒𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → ∃𝑓𝜏′)
28 nfv 1883 . . . . . . . . . . . 12 𝑥 𝑅 FrSe 𝐴
29 nfsbc1v 3488 . . . . . . . . . . . . . 14 𝑥[𝑦 / 𝑥]𝜏
3025, 29nfxfr 1819 . . . . . . . . . . . . 13 𝑥𝜏′
3130nfex 2192 . . . . . . . . . . . 12 𝑥𝑓𝜏′
3228, 31nfan 1868 . . . . . . . . . . 11 𝑥(𝑅 FrSe 𝐴 ∧ ∃𝑓𝜏′)
3330nfeu 2514 . . . . . . . . . . 11 𝑥∃!𝑓𝜏′
3432, 33nfim 1865 . . . . . . . . . 10 𝑥((𝑅 FrSe 𝐴 ∧ ∃𝑓𝜏′) → ∃!𝑓𝜏′)
35 sneq 4220 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦 → {𝑥} = {𝑦})
36 bnj1318 31219 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦 → trCl(𝑥, 𝐴, 𝑅) = trCl(𝑦, 𝐴, 𝑅))
3735, 36uneq12d 3801 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑦 → ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
3837eqeq2d 2661 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) ↔ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
3938anbi2d 740 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ((𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))) ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))))
4021, 22, 23, 24, 25bnj1373 31224 . . . . . . . . . . . . . 14 (𝜏′ ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
4139, 40syl6bbr 278 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → ((𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))) ↔ 𝜏′))
4241exbidv 1890 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (∃𝑓(𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))) ↔ ∃𝑓𝜏′))
4342anbi2d 740 . . . . . . . . . . 11 (𝑥 = 𝑦 → ((𝑅 FrSe 𝐴 ∧ ∃𝑓(𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))) ↔ (𝑅 FrSe 𝐴 ∧ ∃𝑓𝜏′)))
4441eubidv 2518 . . . . . . . . . . 11 (𝑥 = 𝑦 → (∃!𝑓(𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))) ↔ ∃!𝑓𝜏′))
4543, 44imbi12d 333 . . . . . . . . . 10 (𝑥 = 𝑦 → (((𝑅 FrSe 𝐴 ∧ ∃𝑓(𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))) → ∃!𝑓(𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))) ↔ ((𝑅 FrSe 𝐴 ∧ ∃𝑓𝜏′) → ∃!𝑓𝜏′)))
46 biid 251 . . . . . . . . . . 11 ((𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))) ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
4721, 22, 23, 46bnj1321 31221 . . . . . . . . . 10 ((𝑅 FrSe 𝐴 ∧ ∃𝑓(𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))) → ∃!𝑓(𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
4834, 45, 47chvar 2298 . . . . . . . . 9 ((𝑅 FrSe 𝐴 ∧ ∃𝑓𝜏′) → ∃!𝑓𝜏′)
4920, 27, 48syl2anc 694 . . . . . . . 8 ((𝜒𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → ∃!𝑓𝜏′)
5049ex 449 . . . . . . 7 (𝜒 → (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) → ∃!𝑓𝜏′))
5117, 50ralrimi 2986 . . . . . 6 (𝜒 → ∀𝑦 ∈ pred (𝑥, 𝐴, 𝑅)∃!𝑓𝜏′)
52 bnj1489.9 . . . . . . 7 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
5352a1i 11 . . . . . 6 (𝜒𝐻 = {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′}))
5554bnj1366 31026 . . . . . 6 (( pred(𝑥, 𝐴, 𝑅) ∈ V ∧ ∀𝑦 ∈ pred (𝑥, 𝐴, 𝑅)∃!𝑓𝜏′𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}) → 𝐻 ∈ V)
5612, 51, 53, 55syl3anc 1366 . . . . 5 (𝜒𝐻 ∈ V)
57 uniexg 6997 . . . . 5 (𝐻 ∈ V → 𝐻 ∈ V)
5856, 57syl 17 . . . 4 (𝜒 𝐻 ∈ V)
592, 58syl5eqel 2734 . . 3 (𝜒𝑃 ∈ V)
60 snex 4938 . . . 4 {⟨𝑥, (𝐺𝑍)⟩} ∈ V
6160a1i 11 . . 3 (𝜒 → {⟨𝑥, (𝐺𝑍)⟩} ∈ V)
6259, 61bnj1149 30989 . 2 (𝜒 → (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩}) ∈ V)
631, 62syl5eqel 2734 1 (𝜒𝑄 ∈ V)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1054   = wceq 1523  ∃wex 1744   ∈ wcel 2030  ∃!weu 2498  {cab 2637   ≠ wne 2823  ∀wral 2941  ∃wrex 2942  {crab 2945  Vcvv 3231  [wsbc 3468   ∪ cun 3605   ⊆ wss 3607  ∅c0 3948  {csn 4210  ⟨cop 4216  ∪ cuni 4468   class class class wbr 4685  dom cdm 5143   ↾ cres 5145   Fn wfn 5921  ‘cfv 5926   predc-bnj14 30882   Se w-bnj13 30884   FrSe w-bnj15 30886   trClc-bnj18 30888 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991  ax-reg 8538  ax-inf2 8576 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-om 7108  df-1o 7605  df-bnj17 30881  df-bnj14 30883  df-bnj13 30885  df-bnj15 30887  df-bnj18 30889  df-bnj19 30891 This theorem is referenced by:  bnj1312  31252
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