Users' Mathboxes Mathbox for Jonathan Ben-Naim < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  bnj969 Structured version   Visualization version   GIF version

Theorem bnj969 34939
Description: Technical lemma for bnj69 35003. 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
bnj969.1 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj969.2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj969.3 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
bnj969.10 𝐷 = (ω ∖ {∅})
bnj969.12 𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)
bnj969.14 (𝜏 ↔ (𝑓 Fn 𝑛𝜑𝜓))
bnj969.15 (𝜎 ↔ (𝑛𝐷𝑝 = suc 𝑛𝑚𝑛))
Assertion
Ref Expression
bnj969 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝐶 ∈ V)
Distinct variable groups:   𝐴,𝑖,𝑚,𝑦   𝑅,𝑖,𝑚,𝑦   𝑓,𝑖,𝑚,𝑦   𝑖,𝑛,𝑚
Allowed substitution hints:   𝜑(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜓(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜒(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜏(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜎(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐴(𝑓,𝑛,𝑝)   𝐶(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐷(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)   𝑅(𝑓,𝑛,𝑝)   𝑋(𝑦,𝑓,𝑖,𝑚,𝑛,𝑝)

Proof of Theorem bnj969
StepHypRef Expression
1 simpl 482 . . . 4 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → (𝑅 FrSe 𝐴𝑋𝐴))
2 bnj667 34745 . . . . . . 7 ((𝑛𝐷𝑓 Fn 𝑛𝜑𝜓) → (𝑓 Fn 𝑛𝜑𝜓))
3 bnj969.3 . . . . . . 7 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
4 bnj969.14 . . . . . . 7 (𝜏 ↔ (𝑓 Fn 𝑛𝜑𝜓))
52, 3, 43imtr4i 292 . . . . . 6 (𝜒𝜏)
653ad2ant1 1132 . . . . 5 ((𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) → 𝜏)
76adantl 481 . . . 4 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝜏)
83bnj1232 34796 . . . . . . 7 (𝜒𝑛𝐷)
9 vex 3482 . . . . . . . 8 𝑚 ∈ V
109bnj216 34725 . . . . . . 7 (𝑛 = suc 𝑚𝑚𝑛)
11 id 22 . . . . . . 7 (𝑝 = suc 𝑛𝑝 = suc 𝑛)
128, 10, 113anim123i 1150 . . . . . 6 ((𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) → (𝑛𝐷𝑚𝑛𝑝 = suc 𝑛))
13 bnj969.15 . . . . . . 7 (𝜎 ↔ (𝑛𝐷𝑝 = suc 𝑛𝑚𝑛))
14 3ancomb 1098 . . . . . . 7 ((𝑛𝐷𝑝 = suc 𝑛𝑚𝑛) ↔ (𝑛𝐷𝑚𝑛𝑝 = suc 𝑛))
1513, 14bitri 275 . . . . . 6 (𝜎 ↔ (𝑛𝐷𝑚𝑛𝑝 = suc 𝑛))
1612, 15sylibr 234 . . . . 5 ((𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛) → 𝜎)
1716adantl 481 . . . 4 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝜎)
181, 7, 17jca32 515 . . 3 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → ((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜏𝜎)))
19 bnj256 34699 . . 3 ((𝑅 FrSe 𝐴𝑋𝐴𝜏𝜎) ↔ ((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜏𝜎)))
2018, 19sylibr 234 . 2 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → (𝑅 FrSe 𝐴𝑋𝐴𝜏𝜎))
21 bnj969.12 . . 3 𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)
22 bnj969.10 . . . 4 𝐷 = (ω ∖ {∅})
23 bnj969.1 . . . 4 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
24 bnj969.2 . . . 4 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
2522, 4, 13, 23, 24bnj938 34930 . . 3 ((𝑅 FrSe 𝐴𝑋𝐴𝜏𝜎) → 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅) ∈ V)
2621, 25eqeltrid 2843 . 2 ((𝑅 FrSe 𝐴𝑋𝐴𝜏𝜎) → 𝐶 ∈ V)
2720, 26syl 17 1 (((𝑅 FrSe 𝐴𝑋𝐴) ∧ (𝜒𝑛 = suc 𝑚𝑝 = suc 𝑛)) → 𝐶 ∈ V)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wcel 2106  wral 3059  Vcvv 3478  cdif 3960  c0 4339  {csn 4631   ciun 4996  suc csuc 6388   Fn wfn 6558  cfv 6563  ωcom 7887  w-bnj17 34679   predc-bnj14 34681   FrSe w-bnj15 34685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-tr 5266  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fv 6571  df-om 7888  df-bnj17 34680  df-bnj14 34682  df-bnj13 34684  df-bnj15 34686
This theorem is referenced by:  bnj910  34941  bnj1006  34953
  Copyright terms: Public domain W3C validator