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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 5146 | . . . . . . . 8 ⊢ (𝑥 = 𝑋 → (𝑆(𝑅↑𝑟𝑛)𝑥 ↔ 𝑆(𝑅↑𝑟𝑛)𝑋)) | |
4 | 3 | imbi1d 341 | . . . . . . 7 ⊢ (𝑥 = 𝑋 → ((𝑆(𝑅↑𝑟𝑛)𝑥 → 𝜏) ↔ (𝑆(𝑅↑𝑟𝑛)𝑋 → 𝜏))) |
5 | 4 | imbi2d 340 | . . . . . 6 ⊢ (𝑥 = 𝑋 → ((𝑛 ∈ ℕ0 → (𝑆(𝑅↑𝑟𝑛)𝑥 → 𝜏)) ↔ (𝑛 ∈ ℕ0 → (𝑆(𝑅↑𝑟𝑛)𝑋 → 𝜏)))) |
6 | 5 | imbi2d 340 | . . . . 5 ⊢ (𝑥 = 𝑋 → ((𝜂 → (𝑛 ∈ ℕ0 → (𝑆(𝑅↑𝑟𝑛)𝑥 → 𝜏))) ↔ (𝜂 → (𝑛 ∈ ℕ0 → (𝑆(𝑅↑𝑟𝑛)𝑋 → 𝜏))))) |
7 | imbi2 348 | . . . . . . . 8 ⊢ ((𝜓 ↔ 𝜏) → ((𝑆(𝑅↑𝑟𝑛)𝑥 → 𝜓) ↔ (𝑆(𝑅↑𝑟𝑛)𝑥 → 𝜏))) | |
8 | 7 | imbi2d 340 | . . . . . . 7 ⊢ ((𝜓 ↔ 𝜏) → ((𝑛 ∈ ℕ0 → (𝑆(𝑅↑𝑟𝑛)𝑥 → 𝜓)) ↔ (𝑛 ∈ ℕ0 → (𝑆(𝑅↑𝑟𝑛)𝑥 → 𝜏)))) |
9 | 8 | imbi2d 340 | . . . . . 6 ⊢ ((𝜓 ↔ 𝜏) → ((𝜂 → (𝑛 ∈ ℕ0 → (𝑆(𝑅↑𝑟𝑛)𝑥 → 𝜓))) ↔ (𝜂 → (𝑛 ∈ ℕ0 → (𝑆(𝑅↑𝑟𝑛)𝑥 → 𝜏))))) |
10 | 9 | bibi1d 343 | . . . . 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 | imbitrrid 245 | . . . 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 15034 | . . 3 ⊢ (𝜂 → (𝑛 ∈ ℕ0 → (𝑆(𝑅↑𝑟𝑛)𝑥 → 𝜓))) |
21 | 12, 20 | vtoclg 3538 | . 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 1534 ∈ wcel 2099 class class class wbr 5142 Rel wrel 5677 (class class class)co 7414 ℕ0cn0 12494 ↑𝑟crelexp 14990 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11186 ax-resscn 11187 ax-1cn 11188 ax-icn 11189 ax-addcl 11190 ax-addrcl 11191 ax-mulcl 11192 ax-mulrcl 11193 ax-mulcom 11194 ax-addass 11195 ax-mulass 11196 ax-distr 11197 ax-i2m1 11198 ax-1ne0 11199 ax-1rid 11200 ax-rnegex 11201 ax-rrecex 11202 ax-cnre 11203 ax-pre-lttri 11204 ax-pre-lttrn 11205 ax-pre-ltadd 11206 ax-pre-mulgt0 11207 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-riota 7370 df-ov 7417 df-oprab 7418 df-mpo 7419 df-om 7865 df-2nd 7988 df-frecs 8280 df-wrecs 8311 df-recs 8385 df-rdg 8424 df-er 8718 df-en 8956 df-dom 8957 df-sdom 8958 df-pnf 11272 df-mnf 11273 df-xr 11274 df-ltxr 11275 df-le 11276 df-sub 11468 df-neg 11469 df-nn 12235 df-n0 12495 df-z 12581 df-uz 12845 df-seq 13991 df-relexp 14991 |
This theorem is referenced by: rtrclind 15036 |
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