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Theorem bnj1489 31593
Description: Technical lemma for bnj60 31599. 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 31565 . . . . . . . . . 10 (𝑅 FrSe 𝐴𝑅 Se 𝐴)
6 df-bnj13 31229 . . . . . . . . . 10 (𝑅 Se 𝐴 ↔ ∀𝑥𝐴 pred(𝑥, 𝐴, 𝑅) ∈ V)
75, 6sylib 209 . . . . . . . . 9 (𝑅 FrSe 𝐴 → ∀𝑥𝐴 pred(𝑥, 𝐴, 𝑅) ∈ V)
84, 7bnj832 31297 . . . . . . . 8 (𝜓 → ∀𝑥𝐴 pred(𝑥, 𝐴, 𝑅) ∈ V)
93, 8bnj835 31298 . . . . . . 7 (𝜒 → ∀𝑥𝐴 pred(𝑥, 𝐴, 𝑅) ∈ V)
10 bnj1489.5 . . . . . . . 8 𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}
1110, 3bnj1212 31339 . . . . . . 7 (𝜒𝑥𝐴)
129, 11bnj1294 31357 . . . . . 6 (𝜒 → pred(𝑥, 𝐴, 𝑅) ∈ V)
13 nfv 2009 . . . . . . . . 9 𝑦𝜓
14 nfv 2009 . . . . . . . . 9 𝑦 𝑥𝐷
15 nfra1 3088 . . . . . . . . 9 𝑦𝑦𝐷 ¬ 𝑦𝑅𝑥
1613, 14, 15nf3an 2000 . . . . . . . 8 𝑦(𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥)
173, 16nfxfr 1948 . . . . . . 7 𝑦𝜒
184simplbi 491 . . . . . . . . . . 11 (𝜓𝑅 FrSe 𝐴)
193, 18bnj835 31298 . . . . . . . . . 10 (𝜒𝑅 FrSe 𝐴)
2019adantr 472 . . . . . . . . 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 31570 . . . . . . . . . 10 (𝜒 → ∀𝑦 ∈ pred (𝑥, 𝐴, 𝑅)∃𝑓𝜏′)
2726r19.21bi 3079 . . . . . . . . 9 ((𝜒𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → ∃𝑓𝜏′)
28 nfv 2009 . . . . . . . . . . . 12 𝑥 𝑅 FrSe 𝐴
29 nfsbc1v 3618 . . . . . . . . . . . . . 14 𝑥[𝑦 / 𝑥]𝜏
3025, 29nfxfr 1948 . . . . . . . . . . . . 13 𝑥𝜏′
3130nfex 2327 . . . . . . . . . . . 12 𝑥𝑓𝜏′
3228, 31nfan 1998 . . . . . . . . . . 11 𝑥(𝑅 FrSe 𝐴 ∧ ∃𝑓𝜏′)
3330nfeu 2604 . . . . . . . . . . 11 𝑥∃!𝑓𝜏′
3432, 33nfim 1995 . . . . . . . . . 10 𝑥((𝑅 FrSe 𝐴 ∧ ∃𝑓𝜏′) → ∃!𝑓𝜏′)
35 sneq 4346 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦 → {𝑥} = {𝑦})
36 bnj1318 31562 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑦 → trCl(𝑥, 𝐴, 𝑅) = trCl(𝑦, 𝐴, 𝑅))
3735, 36uneq12d 3932 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑦 → ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))
3837eqeq2d 2775 . . . . . . . . . . . . . . 15 (𝑥 = 𝑦 → (dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) ↔ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
3938anbi2d 622 . . . . . . . . . . . . . 14 (𝑥 = 𝑦 → ((𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))) ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)))))
4021, 22, 23, 24, 25bnj1373 31567 . . . . . . . . . . . . . 14 (𝜏′ ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
4139, 40syl6bbr 280 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → ((𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))) ↔ 𝜏′))
4241exbidv 2016 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (∃𝑓(𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))) ↔ ∃𝑓𝜏′))
4342anbi2d 622 . . . . . . . . . . 11 (𝑥 = 𝑦 → ((𝑅 FrSe 𝐴 ∧ ∃𝑓(𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))) ↔ (𝑅 FrSe 𝐴 ∧ ∃𝑓𝜏′)))
4441eubidv 2585 . . . . . . . . . . 11 (𝑥 = 𝑦 → (∃!𝑓(𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))) ↔ ∃!𝑓𝜏′))
4543, 44imbi12d 335 . . . . . . . . . 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 252 . . . . . . . . . . 11 ((𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))) ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
4721, 22, 23, 46bnj1321 31564 . . . . . . . . . 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 2368 . . . . . . . . 9 ((𝑅 FrSe 𝐴 ∧ ∃𝑓𝜏′) → ∃!𝑓𝜏′)
4920, 27, 48syl2anc 579 . . . . . . . 8 ((𝜒𝑦 ∈ pred(𝑥, 𝐴, 𝑅)) → ∃!𝑓𝜏′)
5049ex 401 . . . . . . 7 (𝜒 → (𝑦 ∈ pred(𝑥, 𝐴, 𝑅) → ∃!𝑓𝜏′))
5117, 50ralrimi 3104 . . . . . 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 252 . . . . . . 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 31369 . . . . . 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 1490 . . . . 5 (𝜒𝐻 ∈ V)
57 uniexg 7157 . . . . 5 (𝐻 ∈ V → 𝐻 ∈ V)
5856, 57syl 17 . . . 4 (𝜒 𝐻 ∈ V)
592, 58syl5eqel 2848 . . 3 (𝜒𝑃 ∈ V)
60 snex 5066 . . . 4 {⟨𝑥, (𝐺𝑍)⟩} ∈ V
6160a1i 11 . . 3 (𝜒 → {⟨𝑥, (𝐺𝑍)⟩} ∈ V)
6259, 61bnj1149 31332 . 2 (𝜒 → (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩}) ∈ V)
631, 62syl5eqel 2848 1 (𝜒𝑄 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wex 1874  wcel 2155  ∃!weu 2581  {cab 2751  wne 2937  wral 3055  wrex 3056  {crab 3059  Vcvv 3350  [wsbc 3598  cun 3732  wss 3734  c0 4081  {csn 4336  cop 4342   cuni 4596   class class class wbr 4811  dom cdm 5279  cres 5281   Fn wfn 6065  cfv 6070   predc-bnj14 31226   Se w-bnj13 31228   FrSe w-bnj15 31230   trClc-bnj18 31232
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7151  ax-reg 8708  ax-inf2 8757
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-fal 1666  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-om 7268  df-1o 7768  df-bnj17 31225  df-bnj14 31227  df-bnj13 31229  df-bnj15 31231  df-bnj18 31233  df-bnj19 31235
This theorem is referenced by:  bnj1312  31595
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