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Mirrors > Home > MPE Home > Th. List > uspgrloopvtx | Structured version Visualization version GIF version |
Description: The set of vertices in a graph (simple pseudograph) with one edge which is a loop (see uspgr1v1eop 29049). (Contributed by AV, 17-Dec-2020.) |
Ref | Expression |
---|---|
uspgrloopvtx.g | ⢠ðº = âšð, {âšðŽ, {ð}â©}â© |
Ref | Expression |
---|---|
uspgrloopvtx | ⢠(ð â ð â (Vtxâðº) = ð) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uspgrloopvtx.g | . . 3 ⢠ðº = âšð, {âšðŽ, {ð}â©}â© | |
2 | 1 | fveq2i 6894 | . 2 ⢠(Vtxâðº) = (Vtxââšð, {âšðŽ, {ð}â©}â©) |
3 | snex 5427 | . . 3 ⢠{âšðŽ, {ð}â©} â V | |
4 | opvtxfv 28804 | . . 3 ⢠((ð â ð â§ {âšðŽ, {ð}â©} â V) â (Vtxââšð, {âšðŽ, {ð}â©}â©) = ð) | |
5 | 3, 4 | mpan2 690 | . 2 ⢠(ð â ð â (Vtxââšð, {âšðŽ, {ð}â©}â©) = ð) |
6 | 2, 5 | eqtrid 2779 | 1 ⢠(ð â ð â (Vtxâðº) = ð) |
Colors of variables: wff setvar class |
Syntax hints: â wi 4 = wceq 1534 â wcel 2099 Vcvv 3469 {csn 4624 âšcop 4630 âcfv 6542 Vtxcvtx 28796 |
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 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-iota 6494 df-fun 6544 df-fv 6550 df-1st 7987 df-vtx 28798 |
This theorem is referenced by: uspgrloopvtxel 29317 uspgrloopnb0 29320 uspgrloopvd2 29321 |
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