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Theorem bnj1421 32198
Description: Technical lemma for bnj60 32218. 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 3502 . . . . 5 𝑥 ∈ V
3 fvex 6679 . . . . 5 (𝐺𝑍) ∈ V
42, 3funsn 6403 . . . 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 6070 . . . . . 6 dom {⟨𝑥, (𝐺𝑍)⟩} = {𝑥}
87a1i 11 . . . . 5 (𝜒 → dom {⟨𝑥, (𝐺𝑍)⟩} = {𝑥})
96, 8ineq12d 4193 . . . 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 31916 . . . . . 6 (𝜒𝑅 FrSe 𝐴)
14 biid 262 . . . . . . . 8 (𝑅 FrSe 𝐴𝑅 FrSe 𝐴)
15 biid 262 . . . . . . . 8 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅) ↔ ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))
16 biid 262 . . . . . . . 8 (∀𝑧𝐴 (𝑧𝑅𝑥[𝑧 / 𝑥] ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅)) ↔ ∀𝑧𝐴 (𝑧𝑅𝑥[𝑧 / 𝑥] ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅)))
17 biid 262 . . . . . . . 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 2825 . . . . . . . 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 32197 . . . . . . 7 (𝑅 FrSe 𝐴 → ∀𝑥𝐴 ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))
20 disjsn 4645 . . . . . . . 8 (( trCl(𝑥, 𝐴, 𝑅) ∩ {𝑥}) = ∅ ↔ ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))
2120ralbii 3169 . . . . . . 7 (∀𝑥𝐴 ( trCl(𝑥, 𝐴, 𝑅) ∩ {𝑥}) = ∅ ↔ ∀𝑥𝐴 ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))
2219, 21sylibr 235 . . . . . 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 31957 . . . . 5 (𝜒𝑥𝐴)
2623, 25bnj1294 31975 . . . 4 (𝜒 → ( trCl(𝑥, 𝐴, 𝑅) ∩ {𝑥}) = ∅)
279, 26eqtrd 2860 . . 3 (𝜒 → (dom 𝑃 ∩ dom {⟨𝑥, (𝐺𝑍)⟩}) = ∅)
28 funun 6396 . . 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 6372 . 2 (Fun 𝑄 ↔ Fun (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩}))
3229, 31sylibr 235 1 (𝜒 → Fun 𝑄)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wex 1773  wcel 2107  {cab 2803  wne 3020  wral 3142  wrex 3143  {crab 3146  [wsbc 3775  cun 3937  cin 3938  wss 3939  c0 4294  {csn 4563  cop 4569   cuni 4836   ciun 4916   class class class wbr 5062  dom cdm 5553  cres 5555  Fun wfun 6345   Fn wfn 6346  cfv 6351   predc-bnj14 31844   FrSe w-bnj15 31848   trClc-bnj18 31850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-13 2385  ax-ext 2797  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454  ax-reg 9048  ax-inf2 9096
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-fal 1543  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-pss 3957  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-tp 4568  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-tr 5169  df-id 5458  df-eprel 5463  df-po 5472  df-so 5473  df-fr 5512  df-we 5514  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-ord 6191  df-on 6192  df-lim 6193  df-suc 6194  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-om 7572  df-1o 8096  df-bnj17 31843  df-bnj14 31845  df-bnj13 31847  df-bnj15 31849  df-bnj18 31851  df-bnj19 31853
This theorem is referenced by:  bnj1312  32214
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