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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 6906 | . . . 4 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺)) | |
| 2 | isfusgr.v | . . . 4 ⊢ 𝑉 = (Vtx‘𝐺) | |
| 3 | 1, 2 | eqtr4di 2795 | . . 3 ⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉) |
| 4 | 3 | eleq1d 2826 | . 2 ⊢ (𝑔 = 𝐺 → ((Vtx‘𝑔) ∈ Fin ↔ 𝑉 ∈ Fin)) |
| 5 | df-fusgr 29334 | . 2 ⊢ FinUSGraph = {𝑔 ∈ USGraph ∣ (Vtx‘𝑔) ∈ Fin} | |
| 6 | 4, 5 | elrab2 3695 | 1 ⊢ (𝐺 ∈ FinUSGraph ↔ (𝐺 ∈ USGraph ∧ 𝑉 ∈ Fin)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ‘cfv 6561 Fincfn 8985 Vtxcvtx 29013 USGraphcusgr 29166 FinUSGraphcfusgr 29333 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-iota 6514 df-fv 6569 df-fusgr 29334 |
| This theorem is referenced by: fusgrvtxfi 29336 isfusgrf1 29337 isfusgrcl 29338 fusgrusgr 29339 opfusgr 29340 fusgredgfi 29342 fusgrfis 29347 cusgrsizeindslem 29469 cusgrsizeinds 29470 sizusglecusglem2 29480 fusgrmaxsize 29482 finrusgrfusgr 29583 rusgrnumwwlks 29994 rusgrnumwwlk 29995 frrusgrord0lem 30358 frrusgrord0 30359 clwlknon2num 30387 numclwlk1lem1 30388 numclwlk1lem2 30389 friendshipgt3 30417 |
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