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Theorem bnj1416 32331
Description: Technical lemma for bnj60 32354. 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 5756 . . 3 dom 𝑄 = dom (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩})
3 dmun 5762 . . 3 dom (𝑃 ∪ {⟨𝑥, (𝐺𝑍)⟩}) = (dom 𝑃 ∪ dom {⟨𝑥, (𝐺𝑍)⟩})
4 fvex 6666 . . . . 5 (𝐺𝑍) ∈ V
54dmsnop 6056 . . . 4 dom {⟨𝑥, (𝐺𝑍)⟩} = {𝑥}
65uneq2i 4120 . . 3 (dom 𝑃 ∪ dom {⟨𝑥, (𝐺𝑍)⟩}) = (dom 𝑃 ∪ {𝑥})
72, 3, 63eqtri 2851 . 2 dom 𝑄 = (dom 𝑃 ∪ {𝑥})
8 bnj1416.28 . . . 4 (𝜒 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅))
98uneq1d 4122 . . 3 (𝜒 → (dom 𝑃 ∪ {𝑥}) = ( trCl(𝑥, 𝐴, 𝑅) ∪ {𝑥}))
10 uncom 4113 . . 3 ( trCl(𝑥, 𝐴, 𝑅) ∪ {𝑥}) = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
119, 10syl6eq 2875 . 2 (𝜒 → (dom 𝑃 ∪ {𝑥}) = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
127, 11syl5eq 2871 1 (𝜒 → dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wex 1781  wcel 2115  {cab 2802  wne 3013  wral 3132  wrex 3133  {crab 3136  [wsbc 3757  cun 3916  wss 3918  c0 4274  {csn 4548  cop 4554   cuni 4821   class class class wbr 5049  dom cdm 5538  cres 5540   Fn wfn 6333  cfv 6338   predc-bnj14 31978   FrSe w-bnj15 31982   trClc-bnj18 31984
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5186  ax-nul 5193  ax-pr 5313
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3137  df-rex 3138  df-rab 3141  df-v 3481  df-sbc 3758  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-nul 4275  df-if 4449  df-sn 4549  df-pr 4551  df-op 4555  df-uni 4822  df-br 5050  df-dm 5548  df-iota 6297  df-fv 6346
This theorem is referenced by:  bnj1312  32350
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