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Theorem rtrclind 14424
 Description: Principle of transitive induction. The first four hypotheses give various existences, the next four give necessary substitutions and the last two are the basis and the induction step. (Contributed by Drahflow, 12-Nov-2015.)
Hypotheses
Ref Expression
rtrclind.1 (𝜂 → Rel 𝑅)
rtrclind.2 (𝜂𝑅 ∈ V)
rtrclind.3 (𝜂𝑆 ∈ V)
rtrclind.4 (𝜂𝑋 ∈ V)
rtrclind.5 (𝑖 = 𝑆 → (𝜑𝜒))
rtrclind.6 (𝑖 = 𝑥 → (𝜑𝜓))
rtrclind.7 (𝑖 = 𝑗 → (𝜑𝜃))
rtrclind.8 (𝑥 = 𝑋 → (𝜓𝜏))
rtrclind.9 (𝜂𝜒)
rtrclind.10 (𝜂 → (𝑗𝑅𝑥 → (𝜃𝜓)))
Assertion
Ref Expression
rtrclind (𝜂 → (𝑆(t*‘𝑅)𝑋𝜏))
Distinct variable groups:   𝑥,𝑅,𝑖,𝑗   𝑥,𝑆,𝑖,𝑗   𝑥,𝑋   𝜂,𝑥,𝑖,𝑗   𝜏,𝑥   𝜓,𝑖,𝑗   𝜃,𝑖   𝜑,𝑗,𝑥   𝜒,𝑖
Allowed substitution hints:   𝜑(𝑖)   𝜓(𝑥)   𝜒(𝑥,𝑗)   𝜃(𝑥,𝑗)   𝜏(𝑖,𝑗)   𝑋(𝑖,𝑗)

Proof of Theorem rtrclind
Dummy variable 𝑛 is distinct from all other variables.
StepHypRef Expression
1 rtrclind.1 . . 3 (𝜂 → Rel 𝑅)
2 rtrclind.2 . . 3 (𝜂𝑅 ∈ V)
31, 2dfrtrcl2 14421 . 2 (𝜂 → (t*‘𝑅) = (t*rec‘𝑅))
41, 2dfrtrclrec2 14416 . . . . . 6 (𝜂 → (𝑆(t*rec‘𝑅)𝑋 ↔ ∃𝑛 ∈ ℕ0 𝑆(𝑅𝑟𝑛)𝑋))
54biimpac 482 . . . . 5 ((𝑆(t*rec‘𝑅)𝑋𝜂) → ∃𝑛 ∈ ℕ0 𝑆(𝑅𝑟𝑛)𝑋)
6 simprl 770 . . . . . . . . . 10 ((𝑆(t*rec‘𝑅)𝑋 ∧ (𝜂 ∧ (𝑆(𝑅𝑟𝑛)𝑋𝑛 ∈ ℕ0))) → 𝜂)
7 simprrr 781 . . . . . . . . . 10 ((𝑆(t*rec‘𝑅)𝑋 ∧ (𝜂 ∧ (𝑆(𝑅𝑟𝑛)𝑋𝑛 ∈ ℕ0))) → 𝑛 ∈ ℕ0)
8 simprrl 780 . . . . . . . . . 10 ((𝑆(t*rec‘𝑅)𝑋 ∧ (𝜂 ∧ (𝑆(𝑅𝑟𝑛)𝑋𝑛 ∈ ℕ0))) → 𝑆(𝑅𝑟𝑛)𝑋)
9 rtrclind.3 . . . . . . . . . . 11 (𝜂𝑆 ∈ V)
10 rtrclind.4 . . . . . . . . . . 11 (𝜂𝑋 ∈ V)
11 rtrclind.5 . . . . . . . . . . 11 (𝑖 = 𝑆 → (𝜑𝜒))
12 rtrclind.6 . . . . . . . . . . 11 (𝑖 = 𝑥 → (𝜑𝜓))
13 rtrclind.7 . . . . . . . . . . 11 (𝑖 = 𝑗 → (𝜑𝜃))
14 rtrclind.8 . . . . . . . . . . 11 (𝑥 = 𝑋 → (𝜓𝜏))
15 rtrclind.9 . . . . . . . . . . 11 (𝜂𝜒)
16 rtrclind.10 . . . . . . . . . . 11 (𝜂 → (𝑗𝑅𝑥 → (𝜃𝜓)))
171, 2, 9, 10, 11, 12, 13, 14, 15, 16relexpind 14423 . . . . . . . . . 10 (𝜂 → (𝑛 ∈ ℕ0 → (𝑆(𝑅𝑟𝑛)𝑋𝜏)))
186, 7, 8, 17syl3c 66 . . . . . . . . 9 ((𝑆(t*rec‘𝑅)𝑋 ∧ (𝜂 ∧ (𝑆(𝑅𝑟𝑛)𝑋𝑛 ∈ ℕ0))) → 𝜏)
1918anassrs 471 . . . . . . . 8 (((𝑆(t*rec‘𝑅)𝑋𝜂) ∧ (𝑆(𝑅𝑟𝑛)𝑋𝑛 ∈ ℕ0)) → 𝜏)
2019expcom 417 . . . . . . 7 ((𝑆(𝑅𝑟𝑛)𝑋𝑛 ∈ ℕ0) → ((𝑆(t*rec‘𝑅)𝑋𝜂) → 𝜏))
2120expcom 417 . . . . . 6 (𝑛 ∈ ℕ0 → (𝑆(𝑅𝑟𝑛)𝑋 → ((𝑆(t*rec‘𝑅)𝑋𝜂) → 𝜏)))
2221rexlimiv 3272 . . . . 5 (∃𝑛 ∈ ℕ0 𝑆(𝑅𝑟𝑛)𝑋 → ((𝑆(t*rec‘𝑅)𝑋𝜂) → 𝜏))
235, 22mpcom 38 . . . 4 ((𝑆(t*rec‘𝑅)𝑋𝜂) → 𝜏)
2423expcom 417 . . 3 (𝜂 → (𝑆(t*rec‘𝑅)𝑋𝜏))
25 breq 5054 . . . 4 ((t*‘𝑅) = (t*rec‘𝑅) → (𝑆(t*‘𝑅)𝑋𝑆(t*rec‘𝑅)𝑋))
2625imbi1d 345 . . 3 ((t*‘𝑅) = (t*rec‘𝑅) → ((𝑆(t*‘𝑅)𝑋𝜏) ↔ (𝑆(t*rec‘𝑅)𝑋𝜏)))
2724, 26syl5ibr 249 . 2 ((t*‘𝑅) = (t*rec‘𝑅) → (𝜂 → (𝑆(t*‘𝑅)𝑋𝜏)))
283, 27mpcom 38 1 (𝜂 → (𝑆(t*‘𝑅)𝑋𝜏))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2115  ∃wrex 3134  Vcvv 3480   class class class wbr 5052  Rel wrel 5547  ‘cfv 6343  (class class class)co 7149  ℕ0cn0 11894  t*crtcl 14346  ↑𝑟crelexp 14379  t*reccrtrcl 14414 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5176  ax-sep 5189  ax-nul 5196  ax-pow 5253  ax-pr 5317  ax-un 7455  ax-cnex 10591  ax-resscn 10592  ax-1cn 10593  ax-icn 10594  ax-addcl 10595  ax-addrcl 10596  ax-mulcl 10597  ax-mulrcl 10598  ax-mulcom 10599  ax-addass 10600  ax-mulass 10601  ax-distr 10602  ax-i2m1 10603  ax-1ne0 10604  ax-1rid 10605  ax-rnegex 10606  ax-rrecex 10607  ax-cnre 10608  ax-pre-lttri 10609  ax-pre-lttrn 10610  ax-pre-ltadd 10611  ax-pre-mulgt0 10612 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4825  df-int 4863  df-iun 4907  df-br 5053  df-opab 5115  df-mpt 5133  df-tr 5159  df-id 5447  df-eprel 5452  df-po 5461  df-so 5462  df-fr 5501  df-we 5503  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-res 5554  df-ima 5555  df-pred 6135  df-ord 6181  df-on 6182  df-lim 6183  df-suc 6184  df-iota 6302  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-fv 6351  df-riota 7107  df-ov 7152  df-oprab 7153  df-mpo 7154  df-om 7575  df-2nd 7685  df-wrecs 7943  df-recs 8004  df-rdg 8042  df-er 8285  df-en 8506  df-dom 8507  df-sdom 8508  df-pnf 10675  df-mnf 10676  df-xr 10677  df-ltxr 10678  df-le 10679  df-sub 10870  df-neg 10871  df-nn 11635  df-2 11697  df-n0 11895  df-z 11979  df-uz 12241  df-seq 13374  df-rtrcl 14348  df-relexp 14380  df-rtrclrec 14415 This theorem is referenced by: (None)
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