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Mirrors > Home > MPE Home > Th. List > opfi1ind | Structured version Visualization version GIF version |
Description: Properties of an ordered pair with a finite first component, proven by finite induction on the size of the first component. This theorem can be applied for graphs (represented as orderd pairs of vertices and edges) with a finite number of vertices, e.g. fusgrfis 27040. (Contributed by AV, 22-Oct-2020.) (Revised by AV, 28-Mar-2021.) |
Ref | Expression |
---|---|
opfi1ind.e | ⊢ 𝐸 ∈ V |
opfi1ind.f | ⊢ 𝐹 ∈ V |
opfi1ind.1 | ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝜓 ↔ 𝜑)) |
opfi1ind.2 | ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → (𝜓 ↔ 𝜃)) |
opfi1ind.3 | ⊢ ((〈𝑣, 𝑒〉 ∈ 𝐺 ∧ 𝑛 ∈ 𝑣) → 〈(𝑣 ∖ {𝑛}), 𝐹〉 ∈ 𝐺) |
opfi1ind.4 | ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃 ↔ 𝜒)) |
opfi1ind.base | ⊢ ((〈𝑣, 𝑒〉 ∈ 𝐺 ∧ (♯‘𝑣) = 0) → 𝜓) |
opfi1ind.step | ⊢ ((((𝑦 + 1) ∈ ℕ0 ∧ (〈𝑣, 𝑒〉 ∈ 𝐺 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) ∧ 𝜒) → 𝜓) |
Ref | Expression |
---|---|
opfi1ind | ⊢ ((〈𝑉, 𝐸〉 ∈ 𝐺 ∧ 𝑉 ∈ Fin) → 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hashge0 13738 | . . 3 ⊢ (𝑉 ∈ Fin → 0 ≤ (♯‘𝑉)) | |
2 | 1 | adantl 482 | . 2 ⊢ ((〈𝑉, 𝐸〉 ∈ 𝐺 ∧ 𝑉 ∈ Fin) → 0 ≤ (♯‘𝑉)) |
3 | opfi1ind.e | . . 3 ⊢ 𝐸 ∈ V | |
4 | opfi1ind.f | . . 3 ⊢ 𝐹 ∈ V | |
5 | 0nn0 11901 | . . 3 ⊢ 0 ∈ ℕ0 | |
6 | opfi1ind.1 | . . 3 ⊢ ((𝑣 = 𝑉 ∧ 𝑒 = 𝐸) → (𝜓 ↔ 𝜑)) | |
7 | opfi1ind.2 | . . 3 ⊢ ((𝑣 = 𝑤 ∧ 𝑒 = 𝑓) → (𝜓 ↔ 𝜃)) | |
8 | opfi1ind.3 | . . 3 ⊢ ((〈𝑣, 𝑒〉 ∈ 𝐺 ∧ 𝑛 ∈ 𝑣) → 〈(𝑣 ∖ {𝑛}), 𝐹〉 ∈ 𝐺) | |
9 | opfi1ind.4 | . . 3 ⊢ ((𝑤 = (𝑣 ∖ {𝑛}) ∧ 𝑓 = 𝐹) → (𝜃 ↔ 𝜒)) | |
10 | opfi1ind.base | . . 3 ⊢ ((〈𝑣, 𝑒〉 ∈ 𝐺 ∧ (♯‘𝑣) = 0) → 𝜓) | |
11 | opfi1ind.step | . . 3 ⊢ ((((𝑦 + 1) ∈ ℕ0 ∧ (〈𝑣, 𝑒〉 ∈ 𝐺 ∧ (♯‘𝑣) = (𝑦 + 1) ∧ 𝑛 ∈ 𝑣)) ∧ 𝜒) → 𝜓) | |
12 | 3, 4, 5, 6, 7, 8, 9, 10, 11 | opfi1uzind 13849 | . 2 ⊢ ((〈𝑉, 𝐸〉 ∈ 𝐺 ∧ 𝑉 ∈ Fin ∧ 0 ≤ (♯‘𝑉)) → 𝜑) |
13 | 2, 12 | mpd3an3 1453 | 1 ⊢ ((〈𝑉, 𝐸〉 ∈ 𝐺 ∧ 𝑉 ∈ Fin) → 𝜑) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 Vcvv 3495 ∖ cdif 3932 {csn 4559 〈cop 4565 class class class wbr 5058 ‘cfv 6349 (class class class)co 7145 Fincfn 8498 0cc0 10526 1c1 10527 + caddc 10529 ≤ cle 10665 ℕ0cn0 11886 ♯chash 13680 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-rep 5182 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-cnex 10582 ax-resscn 10583 ax-1cn 10584 ax-icn 10585 ax-addcl 10586 ax-addrcl 10587 ax-mulcl 10588 ax-mulrcl 10589 ax-mulcom 10590 ax-addass 10591 ax-mulass 10592 ax-distr 10593 ax-i2m1 10594 ax-1ne0 10595 ax-1rid 10596 ax-rnegex 10597 ax-rrecex 10598 ax-cnre 10599 ax-pre-lttri 10600 ax-pre-lttrn 10601 ax-pre-ltadd 10602 ax-pre-mulgt0 10603 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-int 4870 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-riota 7103 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7569 df-1st 7680 df-2nd 7681 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-1o 8093 df-oadd 8097 df-er 8279 df-en 8499 df-dom 8500 df-sdom 8501 df-fin 8502 df-dju 9319 df-card 9357 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-sub 10861 df-neg 10862 df-nn 11628 df-n0 11887 df-xnn0 11957 df-z 11971 df-uz 12233 df-fz 12883 df-hash 13681 |
This theorem is referenced by: fusgrfis 27040 cusgrsize 27164 finsumvtxdg2size 27260 |
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