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Mirrors > Home > MPE Home > Th. List > relexpind | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
relexpind.1 | ⊢ (𝜂 → Rel 𝑅) |
relexpind.2 | ⊢ (𝜂 → 𝑆 ∈ 𝑉) |
relexpind.3 | ⊢ (𝜂 → 𝑋 ∈ 𝑊) |
relexpind.4 | ⊢ (𝑖 = 𝑆 → (𝜑 ↔ 𝜒)) |
relexpind.5 | ⊢ (𝑖 = 𝑥 → (𝜑 ↔ 𝜓)) |
relexpind.6 | ⊢ (𝑖 = 𝑗 → (𝜑 ↔ 𝜃)) |
relexpind.7 | ⊢ (𝑥 = 𝑋 → (𝜓 ↔ 𝜏)) |
relexpind.8 | ⊢ (𝜂 → 𝜒) |
relexpind.9 | ⊢ (𝜂 → (𝑗𝑅𝑥 → (𝜃 → 𝜓))) |
Ref | Expression |
---|---|
relexpind | ⊢ (𝜂 → (𝑛 ∈ ℕ0 → (𝑆(𝑅↑𝑟𝑛)𝑋 → 𝜏))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relexpind.3 | . 2 ⊢ (𝜂 → 𝑋 ∈ 𝑊) | |
2 | relexpind.7 | . . . 4 ⊢ (𝑥 = 𝑋 → (𝜓 ↔ 𝜏)) | |
3 | breq2 5108 | . . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑆(𝑅↑𝑟𝑛)𝑥 ↔ 𝑆(𝑅↑𝑟𝑛)𝑋)) | |
4 | 3 | imbi1d 342 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → ((𝑆(𝑅↑𝑟𝑛)𝑥 → 𝜏) ↔ (𝑆(𝑅↑𝑟𝑛)𝑋 → 𝜏))) |
5 | 4 | imbi2d 341 | . . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝑛 ∈ ℕ0 → (𝑆(𝑅↑𝑟𝑛)𝑥 → 𝜏)) ↔ (𝑛 ∈ ℕ0 → (𝑆(𝑅↑𝑟𝑛)𝑋 → 𝜏)))) |
6 | 5 | imbi2d 341 | . . . . 5 ⊢ (𝑥 = 𝑋 → ((𝜂 → (𝑛 ∈ ℕ0 → (𝑆(𝑅↑𝑟𝑛)𝑥 → 𝜏))) ↔ (𝜂 → (𝑛 ∈ ℕ0 → (𝑆(𝑅↑𝑟𝑛)𝑋 → 𝜏))))) |
7 | imbi2 349 | . . . . . . . 8 ⊢ ((𝜓 ↔ 𝜏) → ((𝑆(𝑅↑𝑟𝑛)𝑥 → 𝜓) ↔ (𝑆(𝑅↑𝑟𝑛)𝑥 → 𝜏))) | |
8 | 7 | imbi2d 341 | . . . . . . 7 ⊢ ((𝜓 ↔ 𝜏) → ((𝑛 ∈ ℕ0 → (𝑆(𝑅↑𝑟𝑛)𝑥 → 𝜓)) ↔ (𝑛 ∈ ℕ0 → (𝑆(𝑅↑𝑟𝑛)𝑥 → 𝜏)))) |
9 | 8 | imbi2d 341 | . . . . . 6 ⊢ ((𝜓 ↔ 𝜏) → ((𝜂 → (𝑛 ∈ ℕ0 → (𝑆(𝑅↑𝑟𝑛)𝑥 → 𝜓))) ↔ (𝜂 → (𝑛 ∈ ℕ0 → (𝑆(𝑅↑𝑟𝑛)𝑥 → 𝜏))))) |
10 | 9 | bibi1d 344 | . . . . 5 ⊢ ((𝜓 ↔ 𝜏) → (((𝜂 → (𝑛 ∈ ℕ0 → (𝑆(𝑅↑𝑟𝑛)𝑥 → 𝜓))) ↔ (𝜂 → (𝑛 ∈ ℕ0 → (𝑆(𝑅↑𝑟𝑛)𝑋 → 𝜏)))) ↔ ((𝜂 → (𝑛 ∈ ℕ0 → (𝑆(𝑅↑𝑟𝑛)𝑥 → 𝜏))) ↔ (𝜂 → (𝑛 ∈ ℕ0 → (𝑆(𝑅↑𝑟𝑛)𝑋 → 𝜏)))))) |
11 | 6, 10 | syl5ibr 246 | . . . 4 ⊢ ((𝜓 ↔ 𝜏) → (𝑥 = 𝑋 → ((𝜂 → (𝑛 ∈ ℕ0 → (𝑆(𝑅↑𝑟𝑛)𝑥 → 𝜓))) ↔ (𝜂 → (𝑛 ∈ ℕ0 → (𝑆(𝑅↑𝑟𝑛)𝑋 → 𝜏)))))) |
12 | 2, 11 | mpcom 38 | . . 3 ⊢ (𝑥 = 𝑋 → ((𝜂 → (𝑛 ∈ ℕ0 → (𝑆(𝑅↑𝑟𝑛)𝑥 → 𝜓))) ↔ (𝜂 → (𝑛 ∈ ℕ0 → (𝑆(𝑅↑𝑟𝑛)𝑋 → 𝜏))))) |
13 | relexpind.1 | . . . 4 ⊢ (𝜂 → Rel 𝑅) | |
14 | relexpind.2 | . . . 4 ⊢ (𝜂 → 𝑆 ∈ 𝑉) | |
15 | relexpind.4 | . . . 4 ⊢ (𝑖 = 𝑆 → (𝜑 ↔ 𝜒)) | |
16 | relexpind.5 | . . . 4 ⊢ (𝑖 = 𝑥 → (𝜑 ↔ 𝜓)) | |
17 | relexpind.6 | . . . 4 ⊢ (𝑖 = 𝑗 → (𝜑 ↔ 𝜃)) | |
18 | relexpind.8 | . . . 4 ⊢ (𝜂 → 𝜒) | |
19 | relexpind.9 | . . . 4 ⊢ (𝜂 → (𝑗𝑅𝑥 → (𝜃 → 𝜓))) | |
20 | 13, 14, 15, 16, 17, 18, 19 | relexpindlem 14882 | . . 3 ⊢ (𝜂 → (𝑛 ∈ ℕ0 → (𝑆(𝑅↑𝑟𝑛)𝑥 → 𝜓))) |
21 | 12, 20 | vtoclg 3524 | . 2 ⊢ (𝑋 ∈ 𝑊 → (𝜂 → (𝑛 ∈ ℕ0 → (𝑆(𝑅↑𝑟𝑛)𝑋 → 𝜏)))) |
22 | 1, 21 | mpcom 38 | 1 ⊢ (𝜂 → (𝑛 ∈ ℕ0 → (𝑆(𝑅↑𝑟𝑛)𝑋 → 𝜏))) |
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 ℕ0cn0 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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