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Theorem bnj1421 33022
Description: Technical lemma for bnj60 33042. 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
bnj1421.1 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1421.2 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1421.3 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
bnj1421.4 (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
bnj1421.5 𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}
bnj1421.6 (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))
bnj1421.7 (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))
bnj1421.8 (𝜏′[𝑦 / 𝑥]𝜏)
bnj1421.9 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
bnj1421.10 𝑃 = 𝐻
bnj1421.11 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1421.12 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})
bnj1421.13 (𝜒 → Fun 𝑃)
bnj1421.14 (𝜒 → dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
bnj1421.15 (𝜒 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅))
Assertion
Ref Expression
bnj1421 (𝜒 → Fun 𝑄)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑅
Allowed substitution hints:   𝜓(𝑥,𝑦,𝑓,𝑑)   𝜒(𝑥,𝑦,𝑓,𝑑)   𝜏(𝑥,𝑦,𝑓,𝑑)   𝐴(𝑦,𝑓,𝑑)   𝐵(𝑥,𝑦,𝑓,𝑑)   𝐶(𝑥,𝑦,𝑓,𝑑)   𝐷(𝑥,𝑦,𝑓,𝑑)   𝑃(𝑥,𝑦,𝑓,𝑑)   𝑄(𝑥,𝑦,𝑓,𝑑)   𝑅(𝑦,𝑓,𝑑)   𝐺(𝑥,𝑦,𝑓,𝑑)   𝐻(𝑥,𝑦,𝑓,𝑑)   𝑌(𝑥,𝑦,𝑓,𝑑)   𝑍(𝑥,𝑦,𝑓,𝑑)   𝜏′(𝑥,𝑦,𝑓,𝑑)

Proof of Theorem bnj1421
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 bnj1421.13 . . . 4 (𝜒 → Fun 𝑃)
2 vex 3436 . . . . 5 𝑥 ∈ V
3 fvex 6787 . . . . 5 (𝐺𝑍) ∈ V
42, 3funsn 6487 . . . 4 Fun {⟨𝑥, (𝐺𝑍)⟩}
51, 4jctir 521 . . 3 (𝜒 → (Fun 𝑃 ∧ Fun {⟨𝑥, (𝐺𝑍)⟩}))
6 bnj1421.15 . . . . 5 (𝜒 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅))
73dmsnop 6119 . . . . . 6 dom {⟨𝑥, (𝐺𝑍)⟩} = {𝑥}
87a1i 11 . . . . 5 (𝜒 → dom {⟨𝑥, (𝐺𝑍)⟩} = {𝑥})
96, 8ineq12d 4147 . . . 4 (𝜒 → (dom 𝑃 ∩ dom {⟨𝑥, (𝐺𝑍)⟩}) = ( trCl(𝑥, 𝐴, 𝑅) ∩ {𝑥}))
10 bnj1421.7 . . . . . . 7 (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))
11 bnj1421.6 . . . . . . . 8 (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))
1211simplbi 498 . . . . . . 7 (𝜓𝑅 FrSe 𝐴)
1310, 12bnj835 32739 . . . . . 6 (𝜒𝑅 FrSe 𝐴)
14 biid 260 . . . . . . . 8 (𝑅 FrSe 𝐴𝑅 FrSe 𝐴)
15 biid 260 . . . . . . . 8 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅) ↔ ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))
16 biid 260 . . . . . . . 8 (∀𝑧𝐴 (𝑧𝑅𝑥[𝑧 / 𝑥] ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅)) ↔ ∀𝑧𝐴 (𝑧𝑅𝑥[𝑧 / 𝑥] ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅)))
17 biid 260 . . . . . . . 8 ((𝑅 FrSe 𝐴𝑥𝐴 ∧ ∀𝑧𝐴 (𝑧𝑅𝑥[𝑧 / 𝑥] ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))) ↔ (𝑅 FrSe 𝐴𝑥𝐴 ∧ ∀𝑧𝐴 (𝑧𝑅𝑥[𝑧 / 𝑥] ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))))
18 eqid 2738 . . . . . . . 8 ( pred(𝑥, 𝐴, 𝑅) ∪ 𝑧 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑧, 𝐴, 𝑅)) = ( pred(𝑥, 𝐴, 𝑅) ∪ 𝑧 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑧, 𝐴, 𝑅))
1914, 15, 16, 17, 18bnj1417 33021 . . . . . . 7 (𝑅 FrSe 𝐴 → ∀𝑥𝐴 ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))
20 disjsn 4647 . . . . . . . 8 (( trCl(𝑥, 𝐴, 𝑅) ∩ {𝑥}) = ∅ ↔ ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))
2120ralbii 3092 . . . . . . 7 (∀𝑥𝐴 ( trCl(𝑥, 𝐴, 𝑅) ∩ {𝑥}) = ∅ ↔ ∀𝑥𝐴 ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))
2219, 21sylibr 233 . . . . . 6 (𝑅 FrSe 𝐴 → ∀𝑥𝐴 ( trCl(𝑥, 𝐴, 𝑅) ∩ {𝑥}) = ∅)
2313, 22syl 17 . . . . 5 (𝜒 → ∀𝑥𝐴 ( trCl(𝑥, 𝐴, 𝑅) ∩ {𝑥}) = ∅)
24 bnj1421.5 . . . . . 6 𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}
2524, 10bnj1212 32779 . . . . 5 (𝜒𝑥𝐴)
2623, 25bnj1294 32797 . . . 4 (𝜒 → ( trCl(𝑥, 𝐴, 𝑅) ∩ {𝑥}) = ∅)
279, 26eqtrd 2778 . . 3 (𝜒 → (dom 𝑃 ∩ dom {⟨𝑥, (𝐺𝑍)⟩}) = ∅)
28 funun 6480 . . 3 (((Fun 𝑃 ∧ Fun {⟨𝑥, (𝐺𝑍)⟩}) ∧ (dom 𝑃 ∩ dom {⟨𝑥, (𝐺𝑍)⟩}) = ∅) → Fun (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩}))
295, 27, 28syl2anc 584 . 2 (𝜒 → Fun (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩}))
30 bnj1421.12 . . 3 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})
3130funeqi 6455 . 2 (Fun 𝑄 ↔ Fun (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩}))
3229, 31sylibr 233 1 (𝜒 → Fun 𝑄)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1086   = wceq 1539  wex 1782  wcel 2106  {cab 2715  wne 2943  wral 3064  wrex 3065  {crab 3068  [wsbc 3716  cun 3885  cin 3886  wss 3887  c0 4256  {csn 4561  cop 4567   cuni 4839   ciun 4924   class class class wbr 5074  dom cdm 5589  cres 5591  Fun wfun 6427   Fn wfn 6428  cfv 6433   predc-bnj14 32667   FrSe w-bnj15 32671   trClc-bnj18 32673
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-reg 9351  ax-inf2 9399
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-om 7713  df-1o 8297  df-bnj17 32666  df-bnj14 32668  df-bnj13 32670  df-bnj15 32672  df-bnj18 32674  df-bnj19 32676
This theorem is referenced by:  bnj1312  33038
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