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Theorem bnj1021 31363
Description: Technical lemma for bnj69 31407. 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
bnj1021.1 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj1021.2 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
bnj1021.3 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
bnj1021.4 (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))
bnj1021.5 (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛))
bnj1021.6 (𝜂 ↔ (𝑖𝑛𝑦 ∈ (𝑓𝑖)))
bnj1021.13 𝐷 = (ω ∖ {∅})
bnj1021.14 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
Assertion
Ref Expression
bnj1021 𝑓𝑛𝑖𝑚(𝜃 → (𝜃𝜒𝜂 ∧ ∃𝑝𝜏))
Distinct variable groups:   𝐴,𝑓,𝑖,𝑛,𝑦   𝐷,𝑖   𝑅,𝑓,𝑖,𝑛,𝑦   𝑓,𝑋,𝑖,𝑛,𝑦   𝜒,𝑚,𝑝   𝜂,𝑚,𝑝   𝜃,𝑓,𝑖,𝑛   𝜑,𝑖   𝑚,𝑛,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,𝑛)   𝜃(𝑦,𝑧)   𝜏(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝜂(𝑦,𝑧,𝑓,𝑖,𝑛)   𝐴(𝑧,𝑚,𝑝)   𝐵(𝑦,𝑧,𝑓,𝑖,𝑚,𝑛,𝑝)   𝐷(𝑦,𝑧,𝑓,𝑚,𝑛,𝑝)   𝑅(𝑧,𝑚,𝑝)   𝑋(𝑧,𝑚,𝑝)

Proof of Theorem bnj1021
StepHypRef Expression
1 bnj1021.1 . . . 4 (𝜑 ↔ (𝑓‘∅) = pred(𝑋, 𝐴, 𝑅))
2 bnj1021.2 . . . 4 (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖𝑛 → (𝑓‘suc 𝑖) = 𝑦 ∈ (𝑓𝑖) pred(𝑦, 𝐴, 𝑅)))
3 bnj1021.3 . . . 4 (𝜒 ↔ (𝑛𝐷𝑓 Fn 𝑛𝜑𝜓))
4 bnj1021.4 . . . 4 (𝜃 ↔ (𝑅 FrSe 𝐴𝑋𝐴𝑦 ∈ trCl(𝑋, 𝐴, 𝑅) ∧ 𝑧 ∈ pred(𝑦, 𝐴, 𝑅)))
5 bnj1021.5 . . . 4 (𝜏 ↔ (𝑚 ∈ ω ∧ 𝑛 = suc 𝑚𝑝 = suc 𝑛))
6 bnj1021.6 . . . 4 (𝜂 ↔ (𝑖𝑛𝑦 ∈ (𝑓𝑖)))
7 bnj1021.13 . . . 4 𝐷 = (ω ∖ {∅})
8 bnj1021.14 . . . 4 𝐵 = {𝑓 ∣ ∃𝑛𝐷 (𝑓 Fn 𝑛𝜑𝜓)}
91, 2, 3, 4, 5, 6, 7, 8bnj996 31354 . . 3 𝑓𝑛𝑖𝑚𝑝(𝜃 → (𝜒𝜏𝜂))
10 anclb 571 . . . . . 6 ((𝜃 → (𝜒𝜏𝜂)) ↔ (𝜃 → (𝜃 ∧ (𝜒𝜏𝜂))))
11 bnj252 31100 . . . . . . 7 ((𝜃𝜒𝜏𝜂) ↔ (𝜃 ∧ (𝜒𝜏𝜂)))
1211imbi2i 325 . . . . . 6 ((𝜃 → (𝜃𝜒𝜏𝜂)) ↔ (𝜃 → (𝜃 ∧ (𝜒𝜏𝜂))))
1310, 12bitr4i 267 . . . . 5 ((𝜃 → (𝜒𝜏𝜂)) ↔ (𝜃 → (𝜃𝜒𝜏𝜂)))
14132exbii 1924 . . . 4 (∃𝑚𝑝(𝜃 → (𝜒𝜏𝜂)) ↔ ∃𝑚𝑝(𝜃 → (𝜃𝜒𝜏𝜂)))
15143exbii 1925 . . 3 (∃𝑓𝑛𝑖𝑚𝑝(𝜃 → (𝜒𝜏𝜂)) ↔ ∃𝑓𝑛𝑖𝑚𝑝(𝜃 → (𝜃𝜒𝜏𝜂)))
169, 15mpbi 220 . 2 𝑓𝑛𝑖𝑚𝑝(𝜃 → (𝜃𝜒𝜏𝜂))
17 19.37v 2076 . . . . 5 (∃𝑝(𝜃 → (𝜃𝜒𝜏𝜂)) ↔ (𝜃 → ∃𝑝(𝜃𝜒𝜏𝜂)))
18 bnj1019 31179 . . . . . 6 (∃𝑝(𝜃𝜒𝜏𝜂) ↔ (𝜃𝜒𝜂 ∧ ∃𝑝𝜏))
1918imbi2i 325 . . . . 5 ((𝜃 → ∃𝑝(𝜃𝜒𝜏𝜂)) ↔ (𝜃 → (𝜃𝜒𝜂 ∧ ∃𝑝𝜏)))
2017, 19bitri 264 . . . 4 (∃𝑝(𝜃 → (𝜃𝜒𝜏𝜂)) ↔ (𝜃 → (𝜃𝜒𝜂 ∧ ∃𝑝𝜏)))
21202exbii 1924 . . 3 (∃𝑖𝑚𝑝(𝜃 → (𝜃𝜒𝜏𝜂)) ↔ ∃𝑖𝑚(𝜃 → (𝜃𝜒𝜂 ∧ ∃𝑝𝜏)))
22212exbii 1924 . 2 (∃𝑓𝑛𝑖𝑚𝑝(𝜃 → (𝜃𝜒𝜏𝜂)) ↔ ∃𝑓𝑛𝑖𝑚(𝜃 → (𝜃𝜒𝜂 ∧ ∃𝑝𝜏)))
2316, 22mpbi 220 1 𝑓𝑛𝑖𝑚(𝜃 → (𝜃𝜒𝜂 ∧ ∃𝑝𝜏))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1632  wex 1853  wcel 2140  {cab 2747  wral 3051  wrex 3052  cdif 3713  c0 4059  {csn 4322   ciun 4673  suc csuc 5887   Fn wfn 6045  cfv 6050  ωcom 7232  w-bnj17 31083   predc-bnj14 31085   FrSe w-bnj15 31089   trClc-bnj18 31091
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1989  ax-6 2055  ax-7 2091  ax-8 2142  ax-9 2149  ax-10 2169  ax-11 2184  ax-12 2197  ax-13 2392  ax-ext 2741  ax-sep 4934  ax-nul 4942  ax-pr 5056  ax-un 7116
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2048  df-eu 2612  df-mo 2613  df-clab 2748  df-cleq 2754  df-clel 2757  df-nfc 2892  df-ne 2934  df-ral 3056  df-rex 3057  df-rab 3060  df-v 3343  df-sbc 3578  df-dif 3719  df-un 3721  df-in 3723  df-ss 3730  df-pss 3732  df-nul 4060  df-if 4232  df-pw 4305  df-sn 4323  df-pr 4325  df-tp 4327  df-op 4329  df-uni 4590  df-iun 4675  df-br 4806  df-opab 4866  df-tr 4906  df-eprel 5180  df-po 5188  df-so 5189  df-fr 5226  df-we 5228  df-ord 5888  df-on 5889  df-lim 5890  df-suc 5891  df-fn 6053  df-om 7233  df-bnj17 31084  df-bnj18 31092
This theorem is referenced by:  bnj907  31364
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