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Theorem bnj1018 32224
 Description: Technical lemma for bnj69 32270. 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). See bnj1018g 32223 for a less restrictive version requiring ax-13 2383. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)
Hypotheses
Ref Expression
bnj1018.1 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj1018.2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj1018.3 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
bnj1018.4 (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))
bnj1018.5 (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛))
bnj1018.7 (𝜑′[𝑝 / 𝑛]𝜑)
bnj1018.8 (𝜓′[𝑝 / 𝑛]𝜓)
bnj1018.9 (𝜒′[𝑝 / 𝑛]𝜒)
bnj1018.10 (𝜑″[𝐺 / 𝑓]𝜑′)
bnj1018.11 (𝜓″[𝐺 / 𝑓]𝜓′)
bnj1018.12 (𝜒″[𝐺 / 𝑓]𝜒′)
bnj1018.13 𝐷 = (ω ∖ {∅})
bnj1018.14 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
bnj1018.15 𝐶 = 𝑦 ∈ (𝑓𝑚) pred(𝑦, 𝐴, 𝑅)
bnj1018.16 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
bnj1018.26 (𝜒″ ↔ (𝑝𝐷𝐺 Fn 𝑝𝜑″𝜓″))
bnj1018.29 ((𝜃𝜒𝜏𝜂) → 𝜒″)
bnj1018.30 ((𝜃𝜒𝜏𝜂) → (𝜒″𝑖 ∈ ω ∧ suc 𝑖𝑝))
Assertion
Ref Expression
bnj1018 ((𝜃𝜒𝜂 ∧ ∃𝑝𝜏) → (𝐺‘suc 𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅))
Distinct variable groups:   𝐴,𝑓,𝑖,𝑚,𝑛,𝑦   𝐷,𝑓,𝑖,𝑛   𝑖,𝐺,𝑝   𝑅,𝑓,𝑖,𝑚,𝑛,𝑦   𝑓,𝑋,𝑖,𝑛,𝑦   𝜒,𝑝   𝜂,𝑝   𝑓,𝑝   𝜑,𝑖   𝜃,𝑝   𝑛,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,𝑝)   𝐵(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐶(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐷(𝑦,𝑧,𝑚,𝑝)   𝑅(𝑧,𝑝)   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 bnj1018
StepHypRef Expression
1 df-bnj17 31945 . . 3 ((𝜃𝜒𝜂 ∧ ∃𝑝𝜏) ↔ ((𝜃𝜒𝜂) ∧ ∃𝑝𝜏))
2 bnj258 31966 . . . . . . . 8 ((𝜃𝜒𝜏𝜂) ↔ ((𝜃𝜒𝜂) ∧ 𝜏))
3 bnj1018.29 . . . . . . . 8 ((𝜃𝜒𝜏𝜂) → 𝜒″)
42, 3sylbir 237 . . . . . . 7 (((𝜃𝜒𝜂) ∧ 𝜏) → 𝜒″)
54ex 415 . . . . . 6 ((𝜃𝜒𝜂) → (𝜏𝜒″))
65eximdv 1911 . . . . 5 ((𝜃𝜒𝜂) → (∃𝑝𝜏 → ∃𝑝𝜒″))
7 bnj1018.3 . . . . . 6 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
8 bnj1018.9 . . . . . 6 (𝜒′[𝑝 / 𝑛]𝜒)
9 bnj1018.12 . . . . . 6 (𝜒″[𝐺 / 𝑓]𝜒′)
10 bnj1018.14 . . . . . 6 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
11 bnj1018.16 . . . . . 6 𝐺 = (𝑓 ∪ {⟨𝑛, 𝐶⟩})
127, 8, 9, 10, 11bnj985v 32213 . . . . 5 (𝐺𝐵 ↔ ∃𝑝𝜒″)
136, 12syl6ibr 254 . . . 4 ((𝜃𝜒𝜂) → (∃𝑝𝜏𝐺𝐵))
1413imp 409 . . 3 (((𝜃𝜒𝜂) ∧ ∃𝑝𝜏) → 𝐺𝐵)
151, 14sylbi 219 . 2 ((𝜃𝜒𝜂 ∧ ∃𝑝𝜏) → 𝐺𝐵)
16 bnj1019 32039 . . 3 (∃𝑝(𝜃𝜒𝜏𝜂) ↔ (𝜃𝜒𝜂 ∧ ∃𝑝𝜏))
17 bnj1018.30 . . . . . 6 ((𝜃𝜒𝜏𝜂) → (𝜒″𝑖 ∈ ω ∧ suc 𝑖𝑝))
1817simp3d 1138 . . . . 5 ((𝜃𝜒𝜏𝜂) → suc 𝑖𝑝)
19 bnj1018.26 . . . . . . 7 (𝜒″ ↔ (𝑝𝐷𝐺 Fn 𝑝𝜑″𝜓″))
2019bnj1235 32064 . . . . . 6 (𝜒″𝐺 Fn 𝑝)
21 fndm 6448 . . . . . 6 (𝐺 Fn 𝑝 → dom 𝐺 = 𝑝)
223, 20, 213syl 18 . . . . 5 ((𝜃𝜒𝜏𝜂) → dom 𝐺 = 𝑝)
2318, 22eleqtrrd 2914 . . . 4 ((𝜃𝜒𝜏𝜂) → suc 𝑖 ∈ dom 𝐺)
2423exlimiv 1924 . . 3 (∃𝑝(𝜃𝜒𝜏𝜂) → suc 𝑖 ∈ dom 𝐺)
2516, 24sylbir 237 . 2 ((𝜃𝜒𝜂 ∧ ∃𝑝𝜏) → suc 𝑖 ∈ dom 𝐺)
26 bnj1018.1 . . 3 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
27 bnj1018.2 . . 3 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
28 bnj1018.13 . . 3 𝐷 = (ω ∖ {∅})
2911bnj918 32025 . . 3 𝐺 ∈ V
30 vex 3496 . . . 4 𝑖 ∈ V
3130sucex 7518 . . 3 suc 𝑖 ∈ V
3226, 27, 28, 10, 29, 31bnj1015 32222 . 2 ((𝐺𝐵 ∧ suc 𝑖 ∈ dom 𝐺) → (𝐺‘suc 𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅))
3315, 25, 32syl2anc 586 1 ((𝜃𝜒𝜂 ∧ ∃𝑝𝜏) → (𝐺‘suc 𝑖) ⊆ trCl(𝑋, 𝐴, 𝑅))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1081   = wceq 1530  ∃wex 1773   ∈ wcel 2107  {cab 2797  ∀wral 3136  ∃wrex 3137  Vcvv 3493  [wsbc 3770   ∖ cdif 3931   ∪ cun 3932   ⊆ wss 3934  ∅c0 4289  {csn 4559  ⟨cop 4565  ∪ ciun 4910  dom cdm 5548  suc csuc 6186   Fn wfn 6343  ‘cfv 6348  ωcom 7572   ∧ w-bnj17 31944   predc-bnj14 31946   FrSe w-bnj15 31950   trClc-bnj18 31952 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7453 This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-sbc 3771  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-dm 5558  df-suc 6190  df-iota 6307  df-fn 6351  df-fv 6356  df-bnj17 31945  df-bnj18 31953 This theorem is referenced by:  bnj1020  32225
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