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Mirrors > Home > MPE Home > Th. List > umgr2v2evtx | Structured version Visualization version GIF version |
Description: The set of vertices in a multigraph with two edges connecting the same two vertices. (Contributed by AV, 17-Dec-2020.) |
Ref | Expression |
---|---|
umgr2v2evtx.g | ⢠ðº = âšð, {âš0, {ðŽ, ðµ}â©, âš1, {ðŽ, ðµ}â©}â© |
Ref | Expression |
---|---|
umgr2v2evtx | ⢠(ð â ð â (Vtxâðº) = ð) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | umgr2v2evtx.g | . . 3 ⢠ðº = âšð, {âš0, {ðŽ, ðµ}â©, âš1, {ðŽ, ðµ}â©}â© | |
2 | 1 | fveq2i 6894 | . 2 ⢠(Vtxâðº) = (Vtxââšð, {âš0, {ðŽ, ðµ}â©, âš1, {ðŽ, ðµ}â©}â©) |
3 | prex 5432 | . . 3 ⢠{âš0, {ðŽ, ðµ}â©, âš1, {ðŽ, ðµ}â©} â V | |
4 | opvtxfv 28261 | . . 3 ⢠((ð â ð ⧠{âš0, {ðŽ, ðµ}â©, âš1, {ðŽ, ðµ}â©} â V) â (Vtxââšð, {âš0, {ðŽ, ðµ}â©, âš1, {ðŽ, ðµ}â©}â©) = ð) | |
5 | 3, 4 | mpan2 689 | . 2 ⢠(ð â ð â (Vtxââšð, {âš0, {ðŽ, ðµ}â©, âš1, {ðŽ, ðµ}â©}â©) = ð) |
6 | 2, 5 | eqtrid 2784 | 1 ⢠(ð â ð â (Vtxâðº) = ð) |
Colors of variables: wff setvar class |
Syntax hints: â wi 4 = wceq 1541 â wcel 2106 Vcvv 3474 {cpr 4630 âšcop 4634 âcfv 6543 0cc0 11109 1c1 11110 Vtxcvtx 28253 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7724 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-iota 6495 df-fun 6545 df-fv 6551 df-1st 7974 df-vtx 28255 |
This theorem is referenced by: umgr2v2evtxel 28776 umgr2v2e 28779 umgr2v2enb1 28780 |
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