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Theorem bnj1421 31558
Description: Technical lemma for bnj60 31578. 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 3353 . . . . 5 𝑥 ∈ V
3 fvex 6388 . . . . 5 (𝐺𝑍) ∈ V
42, 3funsn 6120 . . . 4 Fun {⟨𝑥, (𝐺𝑍)⟩}
51, 4jctir 516 . . 3 (𝜒 → (Fun 𝑃 ∧ Fun {⟨𝑥, (𝐺𝑍)⟩}))
6 bnj1421.15 . . . . 5 (𝜒 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅))
73dmsnop 5793 . . . . . 6 dom {⟨𝑥, (𝐺𝑍)⟩} = {𝑥}
87a1i 11 . . . . 5 (𝜒 → dom {⟨𝑥, (𝐺𝑍)⟩} = {𝑥})
96, 8ineq12d 3977 . . . 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 491 . . . . . . 7 (𝜓𝑅 FrSe 𝐴)
1310, 12bnj835 31277 . . . . . 6 (𝜒𝑅 FrSe 𝐴)
14 biid 252 . . . . . . . 8 (𝑅 FrSe 𝐴𝑅 FrSe 𝐴)
15 biid 252 . . . . . . . 8 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅) ↔ ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))
16 biid 252 . . . . . . . 8 (∀𝑧𝐴 (𝑧𝑅𝑥[𝑧 / 𝑥] ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅)) ↔ ∀𝑧𝐴 (𝑧𝑅𝑥[𝑧 / 𝑥] ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅)))
17 biid 252 . . . . . . . 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 2765 . . . . . . . 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 31557 . . . . . . 7 (𝑅 FrSe 𝐴 → ∀𝑥𝐴 ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))
20 disjsn 4402 . . . . . . . 8 (( trCl(𝑥, 𝐴, 𝑅) ∩ {𝑥}) = ∅ ↔ ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))
2120ralbii 3127 . . . . . . 7 (∀𝑥𝐴 ( trCl(𝑥, 𝐴, 𝑅) ∩ {𝑥}) = ∅ ↔ ∀𝑥𝐴 ¬ 𝑥 ∈ trCl(𝑥, 𝐴, 𝑅))
2219, 21sylibr 225 . . . . . 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 31318 . . . . 5 (𝜒𝑥𝐴)
2623, 25bnj1294 31336 . . . 4 (𝜒 → ( trCl(𝑥, 𝐴, 𝑅) ∩ {𝑥}) = ∅)
279, 26eqtrd 2799 . . 3 (𝜒 → (dom 𝑃 ∩ dom {⟨𝑥, (𝐺𝑍)⟩}) = ∅)
28 funun 6113 . . 3 (((Fun 𝑃 ∧ Fun {⟨𝑥, (𝐺𝑍)⟩}) ∧ (dom 𝑃 ∩ dom {⟨𝑥, (𝐺𝑍)⟩}) = ∅) → Fun (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩}))
295, 27, 28syl2anc 579 . 2 (𝜒 → Fun (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩}))
30 bnj1421.12 . . 3 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})
3130funeqi 6089 . 2 (Fun 𝑄 ↔ Fun (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩}))
3229, 31sylibr 225 1 (𝜒 → Fun 𝑄)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  w3a 1107   = wceq 1652  wex 1874  wcel 2155  {cab 2751  wne 2937  wral 3055  wrex 3056  {crab 3059  [wsbc 3596  cun 3730  cin 3731  wss 3732  c0 4079  {csn 4334  cop 4340   cuni 4594   ciun 4676   class class class wbr 4809  dom cdm 5277  cres 5279  Fun wfun 6062   Fn wfn 6063  cfv 6068   predc-bnj14 31205   FrSe w-bnj15 31209   trClc-bnj18 31211
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 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-reg 8704  ax-inf2 8753
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 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-ord 5911  df-on 5912  df-lim 5913  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-om 7264  df-1o 7764  df-bnj17 31204  df-bnj14 31206  df-bnj13 31208  df-bnj15 31210  df-bnj18 31212  df-bnj19 31214
This theorem is referenced by:  bnj1312  31574
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