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Mirrors > Home > MPE Home > Th. List > isfusgr | Structured version Visualization version GIF version |
Description: The property of being a finite simple graph. (Contributed by AV, 3-Jan-2020.) (Revised by AV, 21-Oct-2020.) |
Ref | Expression |
---|---|
isfusgr.v | ⊢ 𝑉 = (Vtx‘𝐺) |
Ref | Expression |
---|---|
isfusgr | ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fveq2 6672 | . . . 4 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺)) | |
2 | isfusgr.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
3 | 1, 2 | syl6eqr 2876 | . . 3 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉) |
4 | 3 | eleq1d 2899 | . 2 ⊢ (𝑔 = 𝐺 → ((Vtx‘𝑔) ∈ Fin ↔ 𝑉 ∈ Fin)) |
5 | df-fusgr 27101 | . 2 ⊢ FinUSGraph = {𝑔 ∈ USGraph ∣ (Vtx‘𝑔) ∈ Fin} | |
6 | 4, 5 | elrab2 3685 | 1 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ‘cfv 6357 Fincfn 8511 Vtxcvtx 26783 USGraphcusgr 26936 FinUSGraphcfusgr 27100 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-iota 6316 df-fv 6365 df-fusgr 27101 |
This theorem is referenced by: fusgrvtxfi 27103 isfusgrf1 27104 isfusgrcl 27105 fusgrusgr 27106 opfusgr 27107 fusgredgfi 27109 fusgrfis 27114 cusgrsizeindslem 27235 cusgrsizeinds 27236 sizusglecusglem2 27246 fusgrmaxsize 27248 finrusgrfusgr 27349 rusgrnumwwlks 27755 rusgrnumwwlk 27756 frrusgrord0lem 28120 frrusgrord0 28121 clwlknon2num 28149 numclwlk1lem1 28150 numclwlk1lem2 28151 friendshipgt3 28179 |
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