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Mirrors > Home > MPE Home > Th. List > usgredgffibi | Structured version Visualization version GIF version |
Description: The number of edges in a simple graph is finite iff its edge function is finite. (Contributed by AV, 10-Jan-2020.) (Revised by AV, 22-Oct-2020.) |
Ref | Expression |
---|---|
usgredgffibi.I | ⊢ 𝐼 = (iEdg‘𝐺) |
usgredgffibi.e | ⊢ 𝐸 = (Edg‘𝐺) |
Ref | Expression |
---|---|
usgredgffibi | ⊢ (𝐺 ∈ USGraph → (𝐸 ∈ Fin ↔ 𝐼 ∈ Fin)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | usgredgffibi.e | . . . 4 ⊢ 𝐸 = (Edg‘𝐺) | |
2 | edgval 28982 | . . . 4 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺) | |
3 | usgredgffibi.I | . . . . . 6 ⊢ 𝐼 = (iEdg‘𝐺) | |
4 | 3 | eqcomi 2735 | . . . . 5 ⊢ (iEdg‘𝐺) = 𝐼 |
5 | 4 | rneqi 5935 | . . . 4 ⊢ ran (iEdg‘𝐺) = ran 𝐼 |
6 | 1, 2, 5 | 3eqtri 2758 | . . 3 ⊢ 𝐸 = ran 𝐼 |
7 | 6 | eleq1i 2817 | . 2 ⊢ (𝐸 ∈ Fin ↔ ran 𝐼 ∈ Fin) |
8 | 3 | fvexi 6907 | . . 3 ⊢ 𝐼 ∈ V |
9 | eqid 2726 | . . . 4 ⊢ (Vtx‘𝐺) = (Vtx‘𝐺) | |
10 | 9, 3 | usgrfs 29090 | . . 3 ⊢ (𝐺 ∈ USGraph → 𝐼:dom 𝐼–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2}) |
11 | f1vrnfibi 9377 | . . 3 ⊢ ((𝐼 ∈ V ∧ 𝐼:dom 𝐼–1-1→{𝑥 ∈ 𝒫 (Vtx‘𝐺) ∣ (♯‘𝑥) = 2}) → (𝐼 ∈ Fin ↔ ran 𝐼 ∈ Fin)) | |
12 | 8, 10, 11 | sylancr 585 | . 2 ⊢ (𝐺 ∈ USGraph → (𝐼 ∈ Fin ↔ ran 𝐼 ∈ Fin)) |
13 | 7, 12 | bitr4id 289 | 1 ⊢ (𝐺 ∈ USGraph → (𝐸 ∈ Fin ↔ 𝐼 ∈ Fin)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1534 ∈ wcel 2099 {crab 3419 Vcvv 3462 𝒫 cpw 4597 dom cdm 5674 ran crn 5675 –1-1→wf1 6543 ‘cfv 6546 Fincfn 8966 2c2 12313 ♯chash 14342 Vtxcvtx 28929 iEdgciedg 28930 Edgcedg 28980 USGraphcusgr 29082 |
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 2167 ax-ext 2697 ax-rep 5282 ax-sep 5296 ax-nul 5303 ax-pow 5361 ax-pr 5425 ax-un 7738 ax-cnex 11205 ax-resscn 11206 ax-1cn 11207 ax-icn 11208 ax-addcl 11209 ax-addrcl 11210 ax-mulcl 11211 ax-mulrcl 11212 ax-mulcom 11213 ax-addass 11214 ax-mulass 11215 ax-distr 11216 ax-i2m1 11217 ax-1ne0 11218 ax-1rid 11219 ax-rnegex 11220 ax-rrecex 11221 ax-cnre 11222 ax-pre-lttri 11223 ax-pre-lttrn 11224 ax-pre-ltadd 11225 ax-pre-mulgt0 11226 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-reu 3365 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3966 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-int 4947 df-iun 4995 df-br 5146 df-opab 5208 df-mpt 5229 df-tr 5263 df-id 5572 df-eprel 5578 df-po 5586 df-so 5587 df-fr 5629 df-we 5631 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-rn 5685 df-res 5686 df-ima 5687 df-pred 6304 df-ord 6371 df-on 6372 df-lim 6373 df-suc 6374 df-iota 6498 df-fun 6548 df-fn 6549 df-f 6550 df-f1 6551 df-fo 6552 df-f1o 6553 df-fv 6554 df-riota 7372 df-ov 7419 df-oprab 7420 df-mpo 7421 df-om 7869 df-1st 7995 df-2nd 7996 df-frecs 8288 df-wrecs 8319 df-recs 8393 df-rdg 8432 df-1o 8488 df-er 8726 df-en 8967 df-dom 8968 df-sdom 8969 df-fin 8970 df-card 9975 df-pnf 11291 df-mnf 11292 df-xr 11293 df-ltxr 11294 df-le 11295 df-sub 11487 df-neg 11488 df-nn 12259 df-2 12321 df-n0 12519 df-z 12605 df-uz 12869 df-fz 13533 df-hash 14343 df-edg 28981 df-usgr 29084 |
This theorem is referenced by: fusgrfisbase 29261 fusgrfisstep 29262 fusgrfis 29263 fusgrfupgrfs 29264 vtxdgfusgrf 29431 |
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