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Mirrors > Home > MPE Home > Th. List > trlreslem | Structured version Visualization version GIF version |
Description: Lemma for trlres 27654. Formerly part of proof of eupthres 28164. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Mario Carneiro, 3-May-2015.) (Revised by AV, 6-Mar-2021.) Hypothesis revised using the prefix operation. (Revised by AV, 30-Nov-2022.) |
Ref | Expression |
---|---|
trlres.v | ⊢ 𝑉 = (Vtx‘𝐺) |
trlres.i | ⊢ 𝐼 = (iEdg‘𝐺) |
trlres.d | ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃) |
trlres.n | ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹))) |
trlres.h | ⊢ 𝐻 = (𝐹 prefix 𝑁) |
Ref | Expression |
---|---|
trlreslem | ⊢ (𝜑 → 𝐻:(0..^(♯‘𝐻))–1-1-onto→dom (𝐼 ↾ (𝐹 “ (0..^𝑁)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | trlres.d | . . . 4 ⊢ (𝜑 → 𝐹(Trails‘𝐺)𝑃) | |
2 | trlres.i | . . . . 5 ⊢ 𝐼 = (iEdg‘𝐺) | |
3 | 2 | trlf1 27652 | . . . 4 ⊢ (𝐹(Trails‘𝐺)𝑃 → 𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼) |
4 | 1, 3 | syl 17 | . . 3 ⊢ (𝜑 → 𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼) |
5 | trlres.n | . . . 4 ⊢ (𝜑 → 𝑁 ∈ (0..^(♯‘𝐹))) | |
6 | elfzouz2 13155 | . . . 4 ⊢ (𝑁 ∈ (0..^(♯‘𝐹)) → (♯‘𝐹) ∈ (ℤ≥‘𝑁)) | |
7 | fzoss2 13168 | . . . 4 ⊢ ((♯‘𝐹) ∈ (ℤ≥‘𝑁) → (0..^𝑁) ⊆ (0..^(♯‘𝐹))) | |
8 | 5, 6, 7 | 3syl 18 | . . 3 ⊢ (𝜑 → (0..^𝑁) ⊆ (0..^(♯‘𝐹))) |
9 | f1ores 6644 | . . 3 ⊢ ((𝐹:(0..^(♯‘𝐹))–1-1→dom 𝐼 ∧ (0..^𝑁) ⊆ (0..^(♯‘𝐹))) → (𝐹 ↾ (0..^𝑁)):(0..^𝑁)–1-1-onto→(𝐹 “ (0..^𝑁))) | |
10 | 4, 8, 9 | syl2anc 587 | . 2 ⊢ (𝜑 → (𝐹 ↾ (0..^𝑁)):(0..^𝑁)–1-1-onto→(𝐹 “ (0..^𝑁))) |
11 | trlres.h | . . . 4 ⊢ 𝐻 = (𝐹 prefix 𝑁) | |
12 | trliswlk 27651 | . . . . . 6 ⊢ (𝐹(Trails‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃) | |
13 | 2 | wlkf 27568 | . . . . . 6 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹 ∈ Word dom 𝐼) |
14 | 1, 12, 13 | 3syl 18 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ Word dom 𝐼) |
15 | fzossfz 13159 | . . . . . 6 ⊢ (0..^(♯‘𝐹)) ⊆ (0...(♯‘𝐹)) | |
16 | 15, 5 | sseldi 3885 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ (0...(♯‘𝐹))) |
17 | pfxres 14142 | . . . . 5 ⊢ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑁 ∈ (0...(♯‘𝐹))) → (𝐹 prefix 𝑁) = (𝐹 ↾ (0..^𝑁))) | |
18 | 14, 16, 17 | syl2anc 587 | . . . 4 ⊢ (𝜑 → (𝐹 prefix 𝑁) = (𝐹 ↾ (0..^𝑁))) |
19 | 11, 18 | syl5eq 2786 | . . 3 ⊢ (𝜑 → 𝐻 = (𝐹 ↾ (0..^𝑁))) |
20 | 11 | fveq2i 6689 | . . . . 5 ⊢ (♯‘𝐻) = (♯‘(𝐹 prefix 𝑁)) |
21 | elfzofz 13156 | . . . . . . 7 ⊢ (𝑁 ∈ (0..^(♯‘𝐹)) → 𝑁 ∈ (0...(♯‘𝐹))) | |
22 | 5, 21 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (0...(♯‘𝐹))) |
23 | pfxlen 14146 | . . . . . 6 ⊢ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑁 ∈ (0...(♯‘𝐹))) → (♯‘(𝐹 prefix 𝑁)) = 𝑁) | |
24 | 14, 22, 23 | syl2anc 587 | . . . . 5 ⊢ (𝜑 → (♯‘(𝐹 prefix 𝑁)) = 𝑁) |
25 | 20, 24 | syl5eq 2786 | . . . 4 ⊢ (𝜑 → (♯‘𝐻) = 𝑁) |
26 | 25 | oveq2d 7198 | . . 3 ⊢ (𝜑 → (0..^(♯‘𝐻)) = (0..^𝑁)) |
27 | wrdf 13972 | . . . . . 6 ⊢ (𝐹 ∈ Word dom 𝐼 → 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼) | |
28 | fimass 6535 | . . . . . 6 ⊢ (𝐹:(0..^(♯‘𝐹))⟶dom 𝐼 → (𝐹 “ (0..^𝑁)) ⊆ dom 𝐼) | |
29 | 13, 27, 28 | 3syl 18 | . . . . 5 ⊢ (𝐹(Walks‘𝐺)𝑃 → (𝐹 “ (0..^𝑁)) ⊆ dom 𝐼) |
30 | 1, 12, 29 | 3syl 18 | . . . 4 ⊢ (𝜑 → (𝐹 “ (0..^𝑁)) ⊆ dom 𝐼) |
31 | ssdmres 5858 | . . . 4 ⊢ ((𝐹 “ (0..^𝑁)) ⊆ dom 𝐼 ↔ dom (𝐼 ↾ (𝐹 “ (0..^𝑁))) = (𝐹 “ (0..^𝑁))) | |
32 | 30, 31 | sylib 221 | . . 3 ⊢ (𝜑 → dom (𝐼 ↾ (𝐹 “ (0..^𝑁))) = (𝐹 “ (0..^𝑁))) |
33 | 19, 26, 32 | f1oeq123d 6624 | . 2 ⊢ (𝜑 → (𝐻:(0..^(♯‘𝐻))–1-1-onto→dom (𝐼 ↾ (𝐹 “ (0..^𝑁))) ↔ (𝐹 ↾ (0..^𝑁)):(0..^𝑁)–1-1-onto→(𝐹 “ (0..^𝑁)))) |
34 | 10, 33 | mpbird 260 | 1 ⊢ (𝜑 → 𝐻:(0..^(♯‘𝐻))–1-1-onto→dom (𝐼 ↾ (𝐹 “ (0..^𝑁)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∈ wcel 2114 ⊆ wss 3853 class class class wbr 5040 dom cdm 5535 ↾ cres 5537 “ cima 5538 ⟶wf 6345 –1-1→wf1 6346 –1-1-onto→wf1o 6348 ‘cfv 6349 (class class class)co 7182 0cc0 10627 ℤ≥cuz 12336 ...cfz 12993 ..^cfzo 13136 ♯chash 13794 Word cword 13967 prefix cpfx 14133 Vtxcvtx 26953 iEdgciedg 26954 Walkscwlks 27550 Trailsctrls 27644 |
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 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5242 ax-pr 5306 ax-un 7491 ax-cnex 10683 ax-resscn 10684 ax-1cn 10685 ax-icn 10686 ax-addcl 10687 ax-addrcl 10688 ax-mulcl 10689 ax-mulrcl 10690 ax-mulcom 10691 ax-addass 10692 ax-mulass 10693 ax-distr 10694 ax-i2m1 10695 ax-1ne0 10696 ax-1rid 10697 ax-rnegex 10698 ax-rrecex 10699 ax-cnre 10700 ax-pre-lttri 10701 ax-pre-lttrn 10702 ax-pre-ltadd 10703 ax-pre-mulgt0 10704 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-ifp 1063 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-int 4847 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6185 df-on 6186 df-lim 6187 df-suc 6188 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7139 df-ov 7185 df-oprab 7186 df-mpo 7187 df-om 7612 df-1st 7726 df-2nd 7727 df-wrecs 7988 df-recs 8049 df-rdg 8087 df-1o 8143 df-er 8332 df-map 8451 df-en 8568 df-dom 8569 df-sdom 8570 df-fin 8571 df-card 9453 df-pnf 10767 df-mnf 10768 df-xr 10769 df-ltxr 10770 df-le 10771 df-sub 10962 df-neg 10963 df-nn 11729 df-n0 11989 df-z 12075 df-uz 12337 df-fz 12994 df-fzo 13137 df-hash 13795 df-word 13968 df-substr 14104 df-pfx 14134 df-wlks 27553 df-trls 27646 |
This theorem is referenced by: trlres 27654 eupthres 28164 |
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