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Mirrors > Home > MPE Home > Th. List > trlreslem | Structured version Visualization version GIF version |
Description: Lemma for trlres 29733. Formerly part of proof of eupthres 30244. (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 29731 | . . . 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 13711 | . . . 4 ⊢ (𝑁 ∈ (0..^(♯‘𝐹)) → (♯‘𝐹) ∈ (ℤ≥‘𝑁)) | |
7 | fzoss2 13724 | . . . 4 ⊢ ((♯‘𝐹) ∈ (ℤ≥‘𝑁) → (0..^𝑁) ⊆ (0..^(♯‘𝐹))) | |
8 | 5, 6, 7 | 3syl 18 | . . 3 ⊢ (𝜑 → (0..^𝑁) ⊆ (0..^(♯‘𝐹))) |
9 | f1ores 6863 | . . 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 584 | . 2 ⊢ (𝜑 → (𝐹 ↾ (0..^𝑁)):(0..^𝑁)–1-1-onto→(𝐹 “ (0..^𝑁))) |
11 | trlres.h | . . . 4 ⊢ 𝐻 = (𝐹 prefix 𝑁) | |
12 | trliswlk 29730 | . . . . . 6 ⊢ (𝐹(Trails‘𝐺)𝑃 → 𝐹(Walks‘𝐺)𝑃) | |
13 | 2 | wlkf 29647 | . . . . . 6 ⊢ (𝐹(Walks‘𝐺)𝑃 → 𝐹 ∈ Word dom 𝐼) |
14 | 1, 12, 13 | 3syl 18 | . . . . 5 ⊢ (𝜑 → 𝐹 ∈ Word dom 𝐼) |
15 | fzossfz 13715 | . . . . . 6 ⊢ (0..^(♯‘𝐹)) ⊆ (0...(♯‘𝐹)) | |
16 | 15, 5 | sselid 3993 | . . . . 5 ⊢ (𝜑 → 𝑁 ∈ (0...(♯‘𝐹))) |
17 | pfxres 14714 | . . . . 5 ⊢ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑁 ∈ (0...(♯‘𝐹))) → (𝐹 prefix 𝑁) = (𝐹 ↾ (0..^𝑁))) | |
18 | 14, 16, 17 | syl2anc 584 | . . . 4 ⊢ (𝜑 → (𝐹 prefix 𝑁) = (𝐹 ↾ (0..^𝑁))) |
19 | 11, 18 | eqtrid 2787 | . . 3 ⊢ (𝜑 → 𝐻 = (𝐹 ↾ (0..^𝑁))) |
20 | 11 | fveq2i 6910 | . . . . 5 ⊢ (♯‘𝐻) = (♯‘(𝐹 prefix 𝑁)) |
21 | elfzofz 13712 | . . . . . . 7 ⊢ (𝑁 ∈ (0..^(♯‘𝐹)) → 𝑁 ∈ (0...(♯‘𝐹))) | |
22 | 5, 21 | syl 17 | . . . . . 6 ⊢ (𝜑 → 𝑁 ∈ (0...(♯‘𝐹))) |
23 | pfxlen 14718 | . . . . . 6 ⊢ ((𝐹 ∈ Word dom 𝐼 ∧ 𝑁 ∈ (0...(♯‘𝐹))) → (♯‘(𝐹 prefix 𝑁)) = 𝑁) | |
24 | 14, 22, 23 | syl2anc 584 | . . . . 5 ⊢ (𝜑 → (♯‘(𝐹 prefix 𝑁)) = 𝑁) |
25 | 20, 24 | eqtrid 2787 | . . . 4 ⊢ (𝜑 → (♯‘𝐻) = 𝑁) |
26 | 25 | oveq2d 7447 | . . 3 ⊢ (𝜑 → (0..^(♯‘𝐻)) = (0..^𝑁)) |
27 | wrdf 14554 | . . . . . 6 ⊢ (𝐹 ∈ Word dom 𝐼 → 𝐹:(0..^(♯‘𝐹))⟶dom 𝐼) | |
28 | fimass 6757 | . . . . . 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 6033 | . . . 4 ⊢ ((𝐹 “ (0..^𝑁)) ⊆ dom 𝐼 ↔ dom (𝐼 ↾ (𝐹 “ (0..^𝑁))) = (𝐹 “ (0..^𝑁))) | |
32 | 30, 31 | sylib 218 | . . 3 ⊢ (𝜑 → dom (𝐼 ↾ (𝐹 “ (0..^𝑁))) = (𝐹 “ (0..^𝑁))) |
33 | 19, 26, 32 | f1oeq123d 6843 | . 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 257 | 1 ⊢ (𝜑 → 𝐻:(0..^(♯‘𝐻))–1-1-onto→dom (𝐼 ↾ (𝐹 “ (0..^𝑁)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ⊆ wss 3963 class class class wbr 5148 dom cdm 5689 ↾ cres 5691 “ cima 5692 ⟶wf 6559 –1-1→wf1 6560 –1-1-onto→wf1o 6562 ‘cfv 6563 (class class class)co 7431 0cc0 11153 ℤ≥cuz 12876 ...cfz 13544 ..^cfzo 13691 ♯chash 14366 Word cword 14549 prefix cpfx 14705 Vtxcvtx 29028 iEdgciedg 29029 Walkscwlks 29629 Trailsctrls 29723 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-ifp 1063 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-fzo 13692 df-hash 14367 df-word 14550 df-substr 14676 df-pfx 14706 df-wlks 29632 df-trls 29725 |
This theorem is referenced by: trlres 29733 eupthres 30244 |
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