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Theorem relexpind 14513
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 5034 . . . . . . . 8 (𝑥 = 𝑋 → (𝑆(𝑅𝑟𝑛)𝑥𝑆(𝑅𝑟𝑛)𝑋))
43imbi1d 345 . . . . . . 7 (𝑥 = 𝑋 → ((𝑆(𝑅𝑟𝑛)𝑥𝜏) ↔ (𝑆(𝑅𝑟𝑛)𝑋𝜏)))
54imbi2d 344 . . . . . 6 (𝑥 = 𝑋 → ((𝑛 ∈ ℕ0 → (𝑆(𝑅𝑟𝑛)𝑥𝜏)) ↔ (𝑛 ∈ ℕ0 → (𝑆(𝑅𝑟𝑛)𝑋𝜏))))
65imbi2d 344 . . . . 5 (𝑥 = 𝑋 → ((𝜂 → (𝑛 ∈ ℕ0 → (𝑆(𝑅𝑟𝑛)𝑥𝜏))) ↔ (𝜂 → (𝑛 ∈ ℕ0 → (𝑆(𝑅𝑟𝑛)𝑋𝜏)))))
7 imbi2 352 . . . . . . . 8 ((𝜓𝜏) → ((𝑆(𝑅𝑟𝑛)𝑥𝜓) ↔ (𝑆(𝑅𝑟𝑛)𝑥𝜏)))
87imbi2d 344 . . . . . . 7 ((𝜓𝜏) → ((𝑛 ∈ ℕ0 → (𝑆(𝑅𝑟𝑛)𝑥𝜓)) ↔ (𝑛 ∈ ℕ0 → (𝑆(𝑅𝑟𝑛)𝑥𝜏))))
98imbi2d 344 . . . . . 6 ((𝜓𝜏) → ((𝜂 → (𝑛 ∈ ℕ0 → (𝑆(𝑅𝑟𝑛)𝑥𝜓))) ↔ (𝜂 → (𝑛 ∈ ℕ0 → (𝑆(𝑅𝑟𝑛)𝑥𝜏)))))
109bibi1d 347 . . . . 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 249 . . . 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 14512 . . 3 (𝜂 → (𝑛 ∈ ℕ0 → (𝑆(𝑅𝑟𝑛)𝑥𝜓)))
2112, 20vtoclg 3470 . 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 209   = wceq 1542  wcel 2114   class class class wbr 5030  Rel wrel 5530  (class class class)co 7170  0cn0 11976  𝑟crelexp 14468
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pow 5232  ax-pr 5296  ax-un 7479  ax-cnex 10671  ax-resscn 10672  ax-1cn 10673  ax-icn 10674  ax-addcl 10675  ax-addrcl 10676  ax-mulcl 10677  ax-mulrcl 10678  ax-mulcom 10679  ax-addass 10680  ax-mulass 10681  ax-distr 10682  ax-i2m1 10683  ax-1ne0 10684  ax-1rid 10685  ax-rnegex 10686  ax-rrecex 10687  ax-cnre 10688  ax-pre-lttri 10689  ax-pre-lttrn 10690  ax-pre-ltadd 10691  ax-pre-mulgt0 10692
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-nel 3039  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3400  df-sbc 3681  df-csb 3791  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-pss 3862  df-nul 4212  df-if 4415  df-pw 4490  df-sn 4517  df-pr 4519  df-tp 4521  df-op 4523  df-uni 4797  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5429  df-eprel 5434  df-po 5442  df-so 5443  df-fr 5483  df-we 5485  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-rn 5536  df-res 5537  df-ima 5538  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6297  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7127  df-ov 7173  df-oprab 7174  df-mpo 7175  df-om 7600  df-2nd 7715  df-wrecs 7976  df-recs 8037  df-rdg 8075  df-er 8320  df-en 8556  df-dom 8557  df-sdom 8558  df-pnf 10755  df-mnf 10756  df-xr 10757  df-ltxr 10758  df-le 10759  df-sub 10950  df-neg 10951  df-nn 11717  df-n0 11977  df-z 12063  df-uz 12325  df-seq 13461  df-relexp 14469
This theorem is referenced by:  rtrclind  14514
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