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Theorem bnj1415 33878
Description: Technical lemma for bnj60 33902. 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
bnj1415.1 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1415.2 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1415.3 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
bnj1415.4 (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
bnj1415.5 𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}
bnj1415.6 (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))
bnj1415.7 (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))
bnj1415.8 (𝜏′[𝑦 / 𝑥]𝜏)
bnj1415.9 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
bnj1415.10 𝑃 = 𝐻
Assertion
Ref Expression
bnj1415 (𝜒 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅))
Distinct variable groups:   𝐴,𝑓,𝑥,𝑦   𝐵,𝑓   𝑦,𝐶   𝑦,𝐷   𝑅,𝑓,𝑥,𝑦   𝑓,𝑑,𝑥   𝜓,𝑦   𝜏,𝑦
Allowed substitution hints:   𝜓(𝑥,𝑓,𝑑)   𝜒(𝑥,𝑦,𝑓,𝑑)   𝜏(𝑥,𝑓,𝑑)   𝐴(𝑑)   𝐵(𝑥,𝑦,𝑑)   𝐶(𝑥,𝑓,𝑑)   𝐷(𝑥,𝑓,𝑑)   𝑃(𝑥,𝑦,𝑓,𝑑)   𝑅(𝑑)   𝐺(𝑥,𝑦,𝑓,𝑑)   𝐻(𝑥,𝑦,𝑓,𝑑)   𝑌(𝑥,𝑦,𝑓,𝑑)   𝜏′(𝑥,𝑦,𝑓,𝑑)

Proof of Theorem bnj1415
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 bnj1415.7 . . . 4 (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))
2 bnj1415.6 . . . . 5 (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))
32simplbi 498 . . . 4 (𝜓𝑅 FrSe 𝐴)
41, 3bnj835 33599 . . 3 (𝜒𝑅 FrSe 𝐴)
5 bnj1415.5 . . . 4 𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}
65, 1bnj1212 33639 . . 3 (𝜒𝑥𝐴)
7 eqid 2731 . . . 4 ( pred(𝑥, 𝐴, 𝑅) ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)) = ( pred(𝑥, 𝐴, 𝑅) ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))
87bnj1414 33877 . . 3 ((𝑅 FrSe 𝐴𝑥𝐴) → trCl(𝑥, 𝐴, 𝑅) = ( pred(𝑥, 𝐴, 𝑅) ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)))
94, 6, 8syl2anc 584 . 2 (𝜒 → trCl(𝑥, 𝐴, 𝑅) = ( pred(𝑥, 𝐴, 𝑅) ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)))
10 iunun 5089 . . . 4 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)) = ( 𝑦 ∈ pred (𝑥, 𝐴, 𝑅){𝑦} ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))
11 iunid 5056 . . . . 5 𝑦 ∈ pred (𝑥, 𝐴, 𝑅){𝑦} = pred(𝑥, 𝐴, 𝑅)
1211uneq1i 4155 . . . 4 ( 𝑦 ∈ pred (𝑥, 𝐴, 𝑅){𝑦} ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)) = ( pred(𝑥, 𝐴, 𝑅) ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))
1310, 12eqtri 2759 . . 3 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)) = ( pred(𝑥, 𝐴, 𝑅) ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅))
14 bnj1415.1 . . . 4 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
15 bnj1415.2 . . . 4 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
16 bnj1415.3 . . . 4 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
17 bnj1415.4 . . . 4 (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
18 bnj1415.8 . . . 4 (𝜏′[𝑦 / 𝑥]𝜏)
19 bnj1415.9 . . . 4 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
20 bnj1415.10 . . . 4 𝑃 = 𝐻
21 biid 260 . . . 4 ((𝜒𝑧 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))) ↔ (𝜒𝑧 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
22 biid 260 . . . 4 (((𝜒𝑧 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))) ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑧 ∈ ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))) ↔ ((𝜒𝑧 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))) ∧ 𝑦 ∈ pred(𝑥, 𝐴, 𝑅) ∧ 𝑧 ∈ ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
2314, 15, 16, 17, 5, 2, 1, 18, 19, 20, 21, 22bnj1398 33874 . . 3 (𝜒 𝑦 ∈ pred (𝑥, 𝐴, 𝑅)({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅)) = dom 𝑃)
2413, 23eqtr3id 2785 . 2 (𝜒 → ( pred(𝑥, 𝐴, 𝑅) ∪ 𝑦 ∈ pred (𝑥, 𝐴, 𝑅) trCl(𝑦, 𝐴, 𝑅)) = dom 𝑃)
259, 24eqtr2d 2772 1 (𝜒 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wex 1781  wcel 2106  {cab 2708  wne 2939  wral 3060  wrex 3069  {crab 3431  [wsbc 3773  cun 3942  wss 3944  c0 4318  {csn 4622  cop 4628   cuni 4901   ciun 4990   class class class wbr 5141  dom cdm 5669  cres 5671   Fn wfn 6527  cfv 6532   predc-bnj14 33528   FrSe w-bnj15 33532   trClc-bnj18 33534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2702  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7708  ax-reg 9569  ax-inf2 9618
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3774  df-csb 3890  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4523  df-pw 4598  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ord 6356  df-on 6357  df-lim 6358  df-suc 6359  df-iota 6484  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-fv 6540  df-om 7839  df-1o 8448  df-bnj17 33527  df-bnj14 33529  df-bnj13 33531  df-bnj15 33533  df-bnj18 33535  df-bnj19 33537
This theorem is referenced by:  bnj1312  33898
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