| Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > edgval | Structured version Visualization version GIF version | ||
| Description: The edges of a graph. (Contributed by AV, 1-Jan-2020.) (Revised by AV, 13-Oct-2020.) (Revised by AV, 8-Dec-2021.) |
| Ref | Expression |
|---|---|
| edgval | ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fveq2 6906 | . . . 4 ⊢ (𝑔 = 𝐺 → (iEdg‘𝑔) = (iEdg‘𝐺)) | |
| 2 | 1 | rneqd 5949 | . . 3 ⊢ (𝑔 = 𝐺 → ran (iEdg‘𝑔) = ran (iEdg‘𝐺)) |
| 3 | df-edg 29065 | . . 3 ⊢ Edg = (𝑔 ∈ V ↦ ran (iEdg‘𝑔)) | |
| 4 | fvex 6919 | . . . 4 ⊢ (iEdg‘𝐺) ∈ V | |
| 5 | 4 | rnex 7932 | . . 3 ⊢ ran (iEdg‘𝐺) ∈ V |
| 6 | 2, 3, 5 | fvmpt 7016 | . 2 ⊢ (𝐺 ∈ V → (Edg‘𝐺) = ran (iEdg‘𝐺)) |
| 7 | rn0 5936 | . . . 4 ⊢ ran ∅ = ∅ | |
| 8 | 7 | a1i 11 | . . 3 ⊢ (¬ 𝐺 ∈ V → ran ∅ = ∅) |
| 9 | fvprc 6898 | . . . 4 ⊢ (¬ 𝐺 ∈ V → (iEdg‘𝐺) = ∅) | |
| 10 | 9 | rneqd 5949 | . . 3 ⊢ (¬ 𝐺 ∈ V → ran (iEdg‘𝐺) = ran ∅) |
| 11 | fvprc 6898 | . . 3 ⊢ (¬ 𝐺 ∈ V → (Edg‘𝐺) = ∅) | |
| 12 | 8, 10, 11 | 3eqtr4rd 2788 | . 2 ⊢ (¬ 𝐺 ∈ V → (Edg‘𝐺) = ran (iEdg‘𝐺)) |
| 13 | 6, 12 | pm2.61i 182 | 1 ⊢ (Edg‘𝐺) = ran (iEdg‘𝐺) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 ran crn 5686 ‘cfv 6561 iEdgciedg 29014 Edgcedg 29064 |
| 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-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| 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-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 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-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-iota 6514 df-fun 6563 df-fv 6569 df-edg 29065 |
| This theorem is referenced by: iedgedg 29067 edgopval 29068 edgstruct 29070 edgiedgb 29071 edg0iedg0 29072 uhgredgn0 29145 upgredgss 29149 umgredgss 29150 edgupgr 29151 uhgrvtxedgiedgb 29153 upgredg 29154 usgredgss 29176 ausgrumgri 29184 ausgrusgri 29185 uspgrf1oedg 29190 uspgrupgrushgr 29196 usgrumgruspgr 29199 usgruspgrb 29200 usgrf1oedg 29224 uhgr2edg 29225 usgrsizedg 29232 usgredg3 29233 ushgredgedg 29246 ushgredgedgloop 29248 usgr1e 29262 edg0usgr 29270 usgr1v0edg 29274 usgrexmpledg 29279 subgrprop3 29293 0grsubgr 29295 0uhgrsubgr 29296 subgruhgredgd 29301 uhgrspansubgrlem 29307 uhgrspan1 29320 upgrres1 29330 usgredgffibi 29341 dfnbgr3 29355 nbupgrres 29381 usgrnbcnvfv 29382 cplgrop 29454 cusgrexi 29460 structtocusgr 29463 cusgrsize 29472 1loopgredg 29519 1egrvtxdg0 29529 umgr2v2eedg 29542 edginwlk 29653 wlkl1loop 29656 wlkvtxedg 29662 uspgr2wlkeq 29664 wlkiswwlks1 29887 wlkiswwlks2lem4 29892 wlkiswwlks2lem5 29893 wlkiswwlks2 29895 wlkiswwlksupgr2 29897 2pthon3v 29963 umgrwwlks2on 29977 clwlkclwwlk 30021 lfuhgr 35123 loop1cycl 35142 dfclnbgr3 47813 isubgredgss 47851 isubgredg 47852 isuspgrim0lem 47871 gricushgr 47886 ushggricedg 47896 stgredg 47923 usgrexmpl1edg 47983 usgrexmpl2edg 47988 gpgedg 48004 |
| Copyright terms: Public domain | W3C validator |