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Theorem bnj1416 35054
Description: Technical lemma for bnj60 35077. 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 5914 . . 3 dom 𝑄 = dom (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})
3 dmun 5920 . . 3 dom (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩}) = (dom 𝑃 ∪ dom {⟨𝑥, (𝐺𝑍)⟩})
4 fvex 6918 . . . . 5 (𝐺𝑍) ∈ V
54dmsnop 6235 . . . 4 dom {⟨𝑥, (𝐺𝑍)⟩} = {𝑥}
65uneq2i 4164 . . 3 (dom 𝑃 ∪ dom {⟨𝑥, (𝐺𝑍)⟩}) = (dom 𝑃 ∪ {𝑥})
72, 3, 63eqtri 2768 . 2 dom 𝑄 = (dom 𝑃 ∪ {𝑥})
8 bnj1416.28 . . . 4 (𝜒 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅))
98uneq1d 4166 . . 3 (𝜒 → (dom 𝑃 ∪ {𝑥}) = ( trCl(𝑥, 𝐴, 𝑅) ∪ {𝑥}))
10 uncom 4157 . . 3 ( trCl(𝑥, 𝐴, 𝑅) ∪ {𝑥}) = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
119, 10eqtrdi 2792 . 2 (𝜒 → (dom 𝑃 ∪ {𝑥}) = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
127, 11eqtrid 2788 1 (𝜒 → dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1086   = wceq 1539  wex 1778  wcel 2107  {cab 2713  wne 2939  wral 3060  wrex 3069  {crab 3435  [wsbc 3787  cun 3948  wss 3950  c0 4332  {csn 4625  cop 4631   cuni 4906   class class class wbr 5142  dom cdm 5684  cres 5686   Fn wfn 6555  cfv 6560   predc-bnj14 34703   FrSe w-bnj15 34707   trClc-bnj18 34709
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-dm 5694  df-iota 6513  df-fv 6568
This theorem is referenced by:  bnj1312  35073
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