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Theorem relexpind 14883
Description: Principle of transitive induction, finite version. The first three hypotheses give various existences, the next four give necessary substitutions and the last two are the basis and the induction hypothesis. (Contributed by Drahflow, 12-Nov-2015.) (Revised by RP, 30-May-2020.) (Revised by AV, 13-Jul-2024.)
Hypotheses
Ref Expression
relexpind.1 (𝜂 → Rel 𝑅)
relexpind.2 (𝜂𝑆𝑉)
relexpind.3 (𝜂𝑋𝑊)
relexpind.4 (𝑖 = 𝑆 → (𝜑𝜒))
relexpind.5 (𝑖 = 𝑥 → (𝜑𝜓))
relexpind.6 (𝑖 = 𝑗 → (𝜑𝜃))
relexpind.7 (𝑥 = 𝑋 → (𝜓𝜏))
relexpind.8 (𝜂𝜒)
relexpind.9 (𝜂 → (𝑗𝑅𝑥 → (𝜃𝜓)))
Assertion
Ref Expression
relexpind (𝜂 → (𝑛 ∈ ℕ0 → (𝑆(𝑅𝑟𝑛)𝑋𝜏)))
Distinct variable groups:   𝑖,𝑗,𝑥,𝑅   𝑆,𝑖,𝑗,𝑥   𝑥,𝑋   𝑥,𝑛   𝜑,𝑗,𝑥   𝜓,𝑖,𝑗   𝜒,𝑖   𝜃,𝑖   𝜏,𝑥   𝜂,𝑖,𝑗,𝑥
Allowed substitution hints:   𝜑(𝑖,𝑛)   𝜓(𝑥,𝑛)   𝜒(𝑥,𝑗,𝑛)   𝜃(𝑥,𝑗,𝑛)   𝜏(𝑖,𝑗,𝑛)   𝜂(𝑛)   𝑅(𝑛)   𝑆(𝑛)   𝑉(𝑥,𝑖,𝑗,𝑛)   𝑊(𝑥,𝑖,𝑗,𝑛)   𝑋(𝑖,𝑗,𝑛)

Proof of Theorem relexpind
StepHypRef Expression
1 relexpind.3 . 2 (𝜂𝑋𝑊)
2 relexpind.7 . . . 4 (𝑥 = 𝑋 → (𝜓𝜏))
3 breq2 5108 . . . . . . . 8 (𝑥 = 𝑋 → (𝑆(𝑅𝑟𝑛)𝑥𝑆(𝑅𝑟𝑛)𝑋))
43imbi1d 342 . . . . . . 7 (𝑥 = 𝑋 → ((𝑆(𝑅𝑟𝑛)𝑥𝜏) ↔ (𝑆(𝑅𝑟𝑛)𝑋𝜏)))
54imbi2d 341 . . . . . 6 (𝑥 = 𝑋 → ((𝑛 ∈ ℕ0 → (𝑆(𝑅𝑟𝑛)𝑥𝜏)) ↔ (𝑛 ∈ ℕ0 → (𝑆(𝑅𝑟𝑛)𝑋𝜏))))
65imbi2d 341 . . . . 5 (𝑥 = 𝑋 → ((𝜂 → (𝑛 ∈ ℕ0 → (𝑆(𝑅𝑟𝑛)𝑥𝜏))) ↔ (𝜂 → (𝑛 ∈ ℕ0 → (𝑆(𝑅𝑟𝑛)𝑋𝜏)))))
7 imbi2 349 . . . . . . . 8 ((𝜓𝜏) → ((𝑆(𝑅𝑟𝑛)𝑥𝜓) ↔ (𝑆(𝑅𝑟𝑛)𝑥𝜏)))
87imbi2d 341 . . . . . . 7 ((𝜓𝜏) → ((𝑛 ∈ ℕ0 → (𝑆(𝑅𝑟𝑛)𝑥𝜓)) ↔ (𝑛 ∈ ℕ0 → (𝑆(𝑅𝑟𝑛)𝑥𝜏))))
98imbi2d 341 . . . . . 6 ((𝜓𝜏) → ((𝜂 → (𝑛 ∈ ℕ0 → (𝑆(𝑅𝑟𝑛)𝑥𝜓))) ↔ (𝜂 → (𝑛 ∈ ℕ0 → (𝑆(𝑅𝑟𝑛)𝑥𝜏)))))
109bibi1d 344 . . . . 5 ((𝜓𝜏) → (((𝜂 → (𝑛 ∈ ℕ0 → (𝑆(𝑅𝑟𝑛)𝑥𝜓))) ↔ (𝜂 → (𝑛 ∈ ℕ0 → (𝑆(𝑅𝑟𝑛)𝑋𝜏)))) ↔ ((𝜂 → (𝑛 ∈ ℕ0 → (𝑆(𝑅𝑟𝑛)𝑥𝜏))) ↔ (𝜂 → (𝑛 ∈ ℕ0 → (𝑆(𝑅𝑟𝑛)𝑋𝜏))))))
116, 10syl5ibr 246 . . . 4 ((𝜓𝜏) → (𝑥 = 𝑋 → ((𝜂 → (𝑛 ∈ ℕ0 → (𝑆(𝑅𝑟𝑛)𝑥𝜓))) ↔ (𝜂 → (𝑛 ∈ ℕ0 → (𝑆(𝑅𝑟𝑛)𝑋𝜏))))))
122, 11mpcom 38 . . 3 (𝑥 = 𝑋 → ((𝜂 → (𝑛 ∈ ℕ0 → (𝑆(𝑅𝑟𝑛)𝑥𝜓))) ↔ (𝜂 → (𝑛 ∈ ℕ0 → (𝑆(𝑅𝑟𝑛)𝑋𝜏)))))
13 relexpind.1 . . . 4 (𝜂 → Rel 𝑅)
14 relexpind.2 . . . 4 (𝜂𝑆𝑉)
15 relexpind.4 . . . 4 (𝑖 = 𝑆 → (𝜑𝜒))
16 relexpind.5 . . . 4 (𝑖 = 𝑥 → (𝜑𝜓))
17 relexpind.6 . . . 4 (𝑖 = 𝑗 → (𝜑𝜃))
18 relexpind.8 . . . 4 (𝜂𝜒)
19 relexpind.9 . . . 4 (𝜂 → (𝑗𝑅𝑥 → (𝜃𝜓)))
2013, 14, 15, 16, 17, 18, 19relexpindlem 14882 . . 3 (𝜂 → (𝑛 ∈ ℕ0 → (𝑆(𝑅𝑟𝑛)𝑥𝜓)))
2112, 20vtoclg 3524 . 2 (𝑋𝑊 → (𝜂 → (𝑛 ∈ ℕ0 → (𝑆(𝑅𝑟𝑛)𝑋𝜏))))
221, 21mpcom 38 1 (𝜂 → (𝑛 ∈ ℕ0 → (𝑆(𝑅𝑟𝑛)𝑋𝜏)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1542  wcel 2107   class class class wbr 5104  Rel wrel 5636  (class class class)co 7350  0cn0 12347  𝑟crelexp 14838
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 2709  ax-sep 5255  ax-nul 5262  ax-pow 5319  ax-pr 5383  ax-un 7663  ax-cnex 11041  ax-resscn 11042  ax-1cn 11043  ax-icn 11044  ax-addcl 11045  ax-addrcl 11046  ax-mulcl 11047  ax-mulrcl 11048  ax-mulcom 11049  ax-addass 11050  ax-mulass 11051  ax-distr 11052  ax-i2m1 11053  ax-1ne0 11054  ax-1rid 11055  ax-rnegex 11056  ax-rrecex 11057  ax-cnre 11058  ax-pre-lttri 11059  ax-pre-lttrn 11060  ax-pre-ltadd 11061  ax-pre-mulgt0 11062
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 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2888  df-ne 2943  df-nel 3049  df-ral 3064  df-rex 3073  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3739  df-csb 3855  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-pss 3928  df-nul 4282  df-if 4486  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4865  df-iun 4955  df-br 5105  df-opab 5167  df-mpt 5188  df-tr 5222  df-id 5529  df-eprel 5535  df-po 5543  df-so 5544  df-fr 5586  df-we 5588  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6250  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6444  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7306  df-ov 7353  df-oprab 7354  df-mpo 7355  df-om 7794  df-2nd 7913  df-frecs 8180  df-wrecs 8211  df-recs 8285  df-rdg 8324  df-er 8582  df-en 8818  df-dom 8819  df-sdom 8820  df-pnf 11125  df-mnf 11126  df-xr 11127  df-ltxr 11128  df-le 11129  df-sub 11321  df-neg 11322  df-nn 12088  df-n0 12348  df-z 12434  df-uz 12697  df-seq 13836  df-relexp 14839
This theorem is referenced by:  rtrclind  14884
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