Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > sizusglecusg | Structured version Visualization version GIF version |
Description: The size of a simple graph with 𝑛 vertices is at most the size of a complete simple graph with 𝑛 vertices (𝑛 may be infinite). (Contributed by Alexander van der Vekens, 13-Jan-2018.) (Revised by AV, 13-Nov-2020.) |
Ref | Expression |
---|---|
fusgrmaxsize.v | ⊢ 𝑉 = (Vtx‘𝐺) |
fusgrmaxsize.e | ⊢ 𝐸 = (Edg‘𝐺) |
usgrsscusgra.h | ⊢ 𝑉 = (Vtx‘𝐻) |
usgrsscusgra.f | ⊢ 𝐹 = (Edg‘𝐻) |
Ref | Expression |
---|---|
sizusglecusg | ⊢ ((𝐺 ∈ USGraph ∧ 𝐻 ∈ ComplUSGraph) → (♯‘𝐸) ≤ (♯‘𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fusgrmaxsize.e | . . . . . . . . 9 ⊢ 𝐸 = (Edg‘𝐺) | |
2 | 1 | fvexi 6677 | . . . . . . . 8 ⊢ 𝐸 ∈ V |
3 | resiexg 7611 | . . . . . . . 8 ⊢ (𝐸 ∈ V → ( I ↾ 𝐸) ∈ V) | |
4 | 2, 3 | mp1i 13 | . . . . . . 7 ⊢ ((𝐺 ∈ USGraph ∧ 𝐻 ∈ ComplUSGraph) → ( I ↾ 𝐸) ∈ V) |
5 | fusgrmaxsize.v | . . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐺) | |
6 | usgrsscusgra.h | . . . . . . . 8 ⊢ 𝑉 = (Vtx‘𝐻) | |
7 | usgrsscusgra.f | . . . . . . . 8 ⊢ 𝐹 = (Edg‘𝐻) | |
8 | 5, 1, 6, 7 | sizusglecusglem1 27235 | . . . . . . 7 ⊢ ((𝐺 ∈ USGraph ∧ 𝐻 ∈ ComplUSGraph) → ( I ↾ 𝐸):𝐸–1-1→𝐹) |
9 | f1eq1 6563 | . . . . . . 7 ⊢ (𝑓 = ( I ↾ 𝐸) → (𝑓:𝐸–1-1→𝐹 ↔ ( I ↾ 𝐸):𝐸–1-1→𝐹)) | |
10 | 4, 8, 9 | spcedv 3597 | . . . . . 6 ⊢ ((𝐺 ∈ USGraph ∧ 𝐻 ∈ ComplUSGraph) → ∃𝑓 𝑓:𝐸–1-1→𝐹) |
11 | 10 | adantl 484 | . . . . 5 ⊢ (((𝐸 ∈ Fin ∧ 𝐹 ∈ Fin) ∧ (𝐺 ∈ USGraph ∧ 𝐻 ∈ ComplUSGraph)) → ∃𝑓 𝑓:𝐸–1-1→𝐹) |
12 | hashdom 13732 | . . . . . . 7 ⊢ ((𝐸 ∈ Fin ∧ 𝐹 ∈ Fin) → ((♯‘𝐸) ≤ (♯‘𝐹) ↔ 𝐸 ≼ 𝐹)) | |
13 | 12 | adantr 483 | . . . . . 6 ⊢ (((𝐸 ∈ Fin ∧ 𝐹 ∈ Fin) ∧ (𝐺 ∈ USGraph ∧ 𝐻 ∈ ComplUSGraph)) → ((♯‘𝐸) ≤ (♯‘𝐹) ↔ 𝐸 ≼ 𝐹)) |
14 | brdomg 8511 | . . . . . . . 8 ⊢ (𝐹 ∈ Fin → (𝐸 ≼ 𝐹 ↔ ∃𝑓 𝑓:𝐸–1-1→𝐹)) | |
15 | 14 | adantl 484 | . . . . . . 7 ⊢ ((𝐸 ∈ Fin ∧ 𝐹 ∈ Fin) → (𝐸 ≼ 𝐹 ↔ ∃𝑓 𝑓:𝐸–1-1→𝐹)) |
16 | 15 | adantr 483 | . . . . . 6 ⊢ (((𝐸 ∈ Fin ∧ 𝐹 ∈ Fin) ∧ (𝐺 ∈ USGraph ∧ 𝐻 ∈ ComplUSGraph)) → (𝐸 ≼ 𝐹 ↔ ∃𝑓 𝑓:𝐸–1-1→𝐹)) |
17 | 13, 16 | bitrd 281 | . . . . 5 ⊢ (((𝐸 ∈ Fin ∧ 𝐹 ∈ Fin) ∧ (𝐺 ∈ USGraph ∧ 𝐻 ∈ ComplUSGraph)) → ((♯‘𝐸) ≤ (♯‘𝐹) ↔ ∃𝑓 𝑓:𝐸–1-1→𝐹)) |
18 | 11, 17 | mpbird 259 | . . . 4 ⊢ (((𝐸 ∈ Fin ∧ 𝐹 ∈ Fin) ∧ (𝐺 ∈ USGraph ∧ 𝐻 ∈ ComplUSGraph)) → (♯‘𝐸) ≤ (♯‘𝐹)) |
19 | 18 | exp31 422 | . . 3 ⊢ (𝐸 ∈ Fin → (𝐹 ∈ Fin → ((𝐺 ∈ USGraph ∧ 𝐻 ∈ ComplUSGraph) → (♯‘𝐸) ≤ (♯‘𝐹)))) |
20 | 5, 1, 6, 7 | sizusglecusglem2 27236 | . . . . . 6 ⊢ ((𝐺 ∈ USGraph ∧ 𝐻 ∈ ComplUSGraph ∧ 𝐹 ∈ Fin) → 𝐸 ∈ Fin) |
21 | 20 | pm2.24d 154 | . . . . 5 ⊢ ((𝐺 ∈ USGraph ∧ 𝐻 ∈ ComplUSGraph ∧ 𝐹 ∈ Fin) → (¬ 𝐸 ∈ Fin → (♯‘𝐸) ≤ (♯‘𝐹))) |
22 | 21 | 3expia 1116 | . . . 4 ⊢ ((𝐺 ∈ USGraph ∧ 𝐻 ∈ ComplUSGraph) → (𝐹 ∈ Fin → (¬ 𝐸 ∈ Fin → (♯‘𝐸) ≤ (♯‘𝐹)))) |
23 | 22 | com13 88 | . . 3 ⊢ (¬ 𝐸 ∈ Fin → (𝐹 ∈ Fin → ((𝐺 ∈ USGraph ∧ 𝐻 ∈ ComplUSGraph) → (♯‘𝐸) ≤ (♯‘𝐹)))) |
24 | 19, 23 | pm2.61i 184 | . 2 ⊢ (𝐹 ∈ Fin → ((𝐺 ∈ USGraph ∧ 𝐻 ∈ ComplUSGraph) → (♯‘𝐸) ≤ (♯‘𝐹))) |
25 | 7 | fvexi 6677 | . . . 4 ⊢ 𝐹 ∈ V |
26 | nfile 13712 | . . . 4 ⊢ ((𝐸 ∈ V ∧ 𝐹 ∈ V ∧ ¬ 𝐹 ∈ Fin) → (♯‘𝐸) ≤ (♯‘𝐹)) | |
27 | 2, 25, 26 | mp3an12 1445 | . . 3 ⊢ (¬ 𝐹 ∈ Fin → (♯‘𝐸) ≤ (♯‘𝐹)) |
28 | 27 | a1d 25 | . 2 ⊢ (¬ 𝐹 ∈ Fin → ((𝐺 ∈ USGraph ∧ 𝐻 ∈ ComplUSGraph) → (♯‘𝐸) ≤ (♯‘𝐹))) |
29 | 24, 28 | pm2.61i 184 | 1 ⊢ ((𝐺 ∈ USGraph ∧ 𝐻 ∈ ComplUSGraph) → (♯‘𝐸) ≤ (♯‘𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∧ w3a 1082 = wceq 1531 ∃wex 1774 ∈ wcel 2108 Vcvv 3493 class class class wbr 5057 I cid 5452 ↾ cres 5550 –1-1→wf1 6345 ‘cfv 6348 ≼ cdom 8499 Fincfn 8501 ≤ cle 10668 ♯chash 13682 Vtxcvtx 26773 Edgcedg 26824 USGraphcusgr 26926 ComplUSGraphccusgr 27184 |
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 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-fal 1544 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-1st 7681 df-2nd 7682 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-2o 8095 df-oadd 8098 df-er 8281 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-dju 9322 df-card 9360 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-nn 11631 df-2 11692 df-n0 11890 df-xnn0 11960 df-z 11974 df-uz 12236 df-fz 12885 df-hash 13683 df-vtx 26775 df-iedg 26776 df-edg 26825 df-uhgr 26835 df-upgr 26859 df-umgr 26860 df-uspgr 26927 df-usgr 26928 df-fusgr 27091 df-nbgr 27107 df-uvtx 27160 df-cplgr 27185 df-cusgr 27186 |
This theorem is referenced by: fusgrmaxsize 27238 |
Copyright terms: Public domain | W3C validator |