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Theorem bnj1416 30815
Description: Technical lemma for bnj60 30838. 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 5285 . . 3 dom 𝑄 = dom (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})
3 dmun 5291 . . 3 dom (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩}) = (dom 𝑃 ∪ dom {⟨𝑥, (𝐺𝑍)⟩})
4 fvex 6158 . . . . 5 (𝐺𝑍) ∈ V
54dmsnop 5568 . . . 4 dom {⟨𝑥, (𝐺𝑍)⟩} = {𝑥}
65uneq2i 3742 . . 3 (dom 𝑃 ∪ dom {⟨𝑥, (𝐺𝑍)⟩}) = (dom 𝑃 ∪ {𝑥})
72, 3, 63eqtri 2647 . 2 dom 𝑄 = (dom 𝑃 ∪ {𝑥})
8 bnj1416.28 . . . 4 (𝜒 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅))
98uneq1d 3744 . . 3 (𝜒 → (dom 𝑃 ∪ {𝑥}) = ( trCl(𝑥, 𝐴, 𝑅) ∪ {𝑥}))
10 uncom 3735 . . 3 ( trCl(𝑥, 𝐴, 𝑅) ∪ {𝑥}) = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
119, 10syl6eq 2671 . 2 (𝜒 → (dom 𝑃 ∪ {𝑥}) = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
127, 11syl5eq 2667 1 (𝜒 → dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  {cab 2607  wne 2790  wral 2907  wrex 2908  {crab 2911  [wsbc 3417  cun 3553  wss 3555  c0 3891  {csn 4148  cop 4154   cuni 4402   class class class wbr 4613  dom cdm 5074  cres 5076   Fn wfn 5842  cfv 5847   predc-bnj14 30461   FrSe w-bnj15 30465   trClc-bnj18 30467
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-dm 5084  df-iota 5810  df-fv 5855
This theorem is referenced by:  bnj1312  30834
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