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Theorem bnj1416 33005
Description: Technical lemma for bnj60 33028. 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
bnj1416.1 𝐵 = {𝑑 ∣ (𝑑𝐴 ∧ ∀𝑥𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1416.2 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1416.3 𝐶 = {𝑓 ∣ ∃𝑑𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥𝑑 (𝑓𝑥) = (𝐺𝑌))}
bnj1416.4 (𝜏 ↔ (𝑓𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
bnj1416.5 𝐷 = {𝑥𝐴 ∣ ¬ ∃𝑓𝜏}
bnj1416.6 (𝜓 ↔ (𝑅 FrSe 𝐴𝐷 ≠ ∅))
bnj1416.7 (𝜒 ↔ (𝜓𝑥𝐷 ∧ ∀𝑦𝐷 ¬ 𝑦𝑅𝑥))
bnj1416.8 (𝜏′[𝑦 / 𝑥]𝜏)
bnj1416.9 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
bnj1416.10 𝑃 = 𝐻
bnj1416.11 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1416.12 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})
bnj1416.28 (𝜒 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅))
Assertion
Ref Expression
bnj1416 (𝜒 → dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))

Proof of Theorem bnj1416
StepHypRef Expression
1 bnj1416.12 . . . 4 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})
21dmeqi 5807 . . 3 dom 𝑄 = dom (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})
3 dmun 5813 . . 3 dom (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩}) = (dom 𝑃 ∪ dom {⟨𝑥, (𝐺𝑍)⟩})
4 fvex 6780 . . . . 5 (𝐺𝑍) ∈ V
54dmsnop 6113 . . . 4 dom {⟨𝑥, (𝐺𝑍)⟩} = {𝑥}
65uneq2i 4094 . . 3 (dom 𝑃 ∪ dom {⟨𝑥, (𝐺𝑍)⟩}) = (dom 𝑃 ∪ {𝑥})
72, 3, 63eqtri 2770 . 2 dom 𝑄 = (dom 𝑃 ∪ {𝑥})
8 bnj1416.28 . . . 4 (𝜒 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅))
98uneq1d 4096 . . 3 (𝜒 → (dom 𝑃 ∪ {𝑥}) = ( trCl(𝑥, 𝐴, 𝑅) ∪ {𝑥}))
10 uncom 4087 . . 3 ( trCl(𝑥, 𝐴, 𝑅) ∪ {𝑥}) = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
119, 10eqtrdi 2794 . 2 (𝜒 → (dom 𝑃 ∪ {𝑥}) = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
127, 11eqtrid 2790 1 (𝜒 → dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
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  wss 3887  c0 4257  {csn 4562  cop 4568   cuni 4840   class class class wbr 5074  dom cdm 5585  cres 5587   Fn wfn 6422  cfv 6427   predc-bnj14 32653   FrSe w-bnj15 32657   trClc-bnj18 32659
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-sep 5222  ax-nul 5229  ax-pr 5351
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  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-ral 3069  df-rex 3070  df-rab 3073  df-v 3432  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4258  df-if 4461  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4841  df-br 5075  df-dm 5595  df-iota 6385  df-fv 6435
This theorem is referenced by:  bnj1312  33024
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