Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj96 Structured version   Visualization version   GIF version

Theorem bnj96 34879
Description: Technical lemma for bnj150 34890. 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.) (Revised by Mario Carneiro, 6-May-2015.) (New usage is discouraged.)
Hypothesis
Ref Expression
bnj96.1 𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}
Assertion
Ref Expression
bnj96 ((𝑅 FrSe 𝐴𝑥𝐴) → dom 𝐹 = 1o)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑅
Allowed substitution hint:   𝐹(𝑥)

Proof of Theorem bnj96
StepHypRef Expression
1 bnj93 34877 . . 3 ((𝑅 FrSe 𝐴𝑥𝐴) → pred(𝑥, 𝐴, 𝑅) ∈ V)
2 dmsnopg 6233 . . 3 ( pred(𝑥, 𝐴, 𝑅) ∈ V → dom {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} = {∅})
31, 2syl 17 . 2 ((𝑅 FrSe 𝐴𝑥𝐴) → dom {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩} = {∅})
4 bnj96.1 . . 3 𝐹 = {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}
54dmeqi 5915 . 2 dom 𝐹 = dom {⟨∅, pred(𝑥, 𝐴, 𝑅)⟩}
6 df1o2 8513 . 2 1o = {∅}
73, 5, 63eqtr4g 2802 1 ((𝑅 FrSe 𝐴𝑥𝐴) → dom 𝐹 = 1o)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  c0 4333  {csn 4626  cop 4632  dom cdm 5685  1oc1o 8499   predc-bnj14 34702   FrSe w-bnj15 34706
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-dm 5695  df-suc 6390  df-1o 8506  df-bnj13 34705  df-bnj15 34707
This theorem is referenced by:  bnj150  34890
  Copyright terms: Public domain W3C validator