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Theorem bnj1416 31445
Description: Technical lemma for bnj60 31468. 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 5463 . . 3 dom 𝑄 = dom (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})
3 dmun 5469 . . 3 dom (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩}) = (dom 𝑃 ∪ dom {⟨𝑥, (𝐺𝑍)⟩})
4 fvex 6342 . . . . 5 (𝐺𝑍) ∈ V
54dmsnop 5751 . . . 4 dom {⟨𝑥, (𝐺𝑍)⟩} = {𝑥}
65uneq2i 3915 . . 3 (dom 𝑃 ∪ dom {⟨𝑥, (𝐺𝑍)⟩}) = (dom 𝑃 ∪ {𝑥})
72, 3, 63eqtri 2797 . 2 dom 𝑄 = (dom 𝑃 ∪ {𝑥})
8 bnj1416.28 . . . 4 (𝜒 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅))
98uneq1d 3917 . . 3 (𝜒 → (dom 𝑃 ∪ {𝑥}) = ( trCl(𝑥, 𝐴, 𝑅) ∪ {𝑥}))
10 uncom 3908 . . 3 ( trCl(𝑥, 𝐴, 𝑅) ∪ {𝑥}) = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
119, 10syl6eq 2821 . 2 (𝜒 → (dom 𝑃 ∪ {𝑥}) = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
127, 11syl5eq 2817 1 (𝜒 → dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wex 1852  wcel 2145  {cab 2757  wne 2943  wral 3061  wrex 3062  {crab 3065  [wsbc 3587  cun 3721  wss 3723  c0 4063  {csn 4316  cop 4322   cuni 4574   class class class wbr 4786  dom cdm 5249  cres 5251   Fn wfn 6026  cfv 6031   predc-bnj14 31094   FrSe w-bnj15 31098   trClc-bnj18 31100
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-dm 5259  df-iota 5994  df-fv 6039
This theorem is referenced by:  bnj1312  31464
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