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Theorem bnj1020 33634
Description: Technical lemma for bnj69 33679. 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
bnj1020.1 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj1020.2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj1020.3 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
bnj1020.4 (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))
bnj1020.5 (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛))
bnj1020.6 (𝜂 ↔ (𝑖𝑛𝑦 ∈ (𝑓𝑖)))
bnj1020.7 (𝜑′[𝑝 / 𝑛]𝜑)
bnj1020.8 (𝜓′[𝑝 / 𝑛]𝜓)
bnj1020.9 (𝜒′[𝑝 / 𝑛]𝜒)
bnj1020.10 (𝜑″[𝐺 / 𝑓]𝜑′)
bnj1020.11 (𝜓″[𝐺 / 𝑓]𝜓′)
bnj1020.12 (𝜒″[𝐺 / 𝑓]𝜒′)
bnj1020.13 𝐷 = (ω ∖ {∅})
bnj1020.14 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
bnj1020.15 𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)
bnj1020.16 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
bnj1020.26 (𝜒″ ↔ (𝑝𝐷𝐺 Fn 𝑝𝜑″𝜓″))
Assertion
Ref Expression
bnj1020 ((𝜃𝜒𝜂 ∧ ∃𝑝𝜏) → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
Distinct variable groups:   𝐴,𝑓,𝑖,𝑚,𝑛,𝑦   𝐴,𝑝,𝑓,𝑖,𝑛,𝑦   𝐷,𝑓,𝑖,𝑛   𝑖,𝐺,𝑝   𝑅,𝑓,𝑖,𝑚,𝑛,𝑦   𝑅,𝑝   𝑓,𝑋,𝑖,𝑛,𝑦   𝜒,𝑝   𝜂,𝑝   𝜑,𝑖   𝜃,𝑝
Allowed substitution hints:   𝜑(𝑦,𝑧,𝑓,𝑚,𝑛,𝑝)   𝜓(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜒(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛)   𝜃(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛)   𝜏(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜂(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛)   𝐴(𝑧)   𝐵(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐶(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐷(𝑦,𝑧,𝑚,𝑝)   𝑅(𝑧)   𝐺(𝑦,𝑧,𝑓,𝑚,𝑛)   𝑋(𝑧,𝑚,𝑝)   𝜑′(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜓′(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜒′(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜑″(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜓″(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜒″(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)

Proof of Theorem bnj1020
StepHypRef Expression
1 bnj1019 33448 . . 3 (∃𝑝(𝜃𝜒𝜏𝜂) ↔ (𝜃𝜒𝜂 ∧ ∃𝑝𝜏))
2 bnj1020.1 . . . . 5 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
3 bnj1020.2 . . . . 5 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
4 bnj1020.3 . . . . 5 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
5 bnj1020.4 . . . . 5 (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))
6 bnj1020.5 . . . . 5 (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛))
7 bnj1020.6 . . . . 5 (𝜂 ↔ (𝑖𝑛𝑦 ∈ (𝑓𝑖)))
8 bnj1020.7 . . . . 5 (𝜑′[𝑝 / 𝑛]𝜑)
9 bnj1020.8 . . . . 5 (𝜓′[𝑝 / 𝑛]𝜓)
10 bnj1020.9 . . . . 5 (𝜒′[𝑝 / 𝑛]𝜒)
11 bnj1020.10 . . . . 5 (𝜑″[𝐺 / 𝑓]𝜑′)
12 bnj1020.11 . . . . 5 (𝜓″[𝐺 / 𝑓]𝜓′)
13 bnj1020.12 . . . . 5 (𝜒″[𝐺 / 𝑓]𝜒′)
14 bnj1020.13 . . . . 5 𝐷 = (ω ∖ {∅})
15 bnj1020.15 . . . . 5 𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)
16 bnj1020.16 . . . . 5 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
17 bnj1020.14 . . . . . . 7 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
182, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 17, 15, 16bnj998 33626 . . . . . 6 ((𝜃𝜒𝜏𝜂) → 𝜒″)
194, 6, 7, 14, 18bnj1001 33628 . . . . 5 ((𝜃𝜒𝜏𝜂) → (𝜒″𝑖 ∈ ω ∧ suc 𝑖𝑝))
202, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 19bnj1006 33629 . . . 4 ((𝜃𝜒𝜏𝜂) → pred(𝑦, 𝐴, 𝑅) ⊆ (𝐺‘suc 𝑖))
2120exlimiv 1934 . . 3 (∃𝑝(𝜃𝜒𝜏𝜂) → pred(𝑦, 𝐴, 𝑅) ⊆ (𝐺‘suc 𝑖))
221, 21sylbir 234 . 2 ((𝜃𝜒𝜂 ∧ ∃𝑝𝜏) → pred(𝑦, 𝐴, 𝑅) ⊆ (𝐺‘suc 𝑖))
23 bnj1020.26 . . 3 (𝜒″ ↔ (𝑝𝐷𝐺 Fn 𝑝𝜑″𝜓″))
242, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 17, 15, 16, 23, 18, 19bnj1018 33633 . 2 ((𝜃𝜒𝜂 ∧ ∃𝑝𝜏) → (𝐺‘suc 𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅))
2522, 24sstrd 3955 1 ((𝜃𝜒𝜂 ∧ ∃𝑝𝜏) → pred(𝑦, 𝐴, 𝑅) ⊆ trCl(𝑋, 𝐴, 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  w3a 1088   = wceq 1542  wex 1782  wcel 2107  {cab 2710  wral 3061  wrex 3070  [wsbc 3740  cdif 3908  cun 3909  wss 3911  c0 4283  {csn 4587  cop 4593   ciun 4955  suc csuc 6320   Fn wfn 6492  cfv 6497  ωcom 7803  w-bnj17 33355   predc-bnj14 33357   FrSe w-bnj15 33361   trClc-bnj18 33363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673  ax-reg 9533
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-res 5646  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-fv 6505  df-om 7804  df-bnj17 33356  df-bnj14 33358  df-bnj13 33360  df-bnj15 33362  df-bnj18 33364
This theorem is referenced by:  bnj907  33636
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