![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > opfusgr | Structured version Visualization version GIF version |
Description: A finite simple graph represented as ordered pair. (Contributed by AV, 23-Oct-2020.) |
Ref | Expression |
---|---|
opfusgr | ⢠((ð â ð ⧠ðž â ð) â (âšð, ðžâ© â FinUSGraph â (âšð, ðžâ© â USGraph ⧠ð â Fin))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2733 | . . 3 ⢠(Vtxââšð, ðžâ©) = (Vtxââšð, ðžâ©) | |
2 | 1 | isfusgr 28575 | . 2 ⢠(âšð, ðžâ© â FinUSGraph â (âšð, ðžâ© â USGraph ⧠(Vtxââšð, ðžâ©) â Fin)) |
3 | opvtxfv 28264 | . . . 4 ⢠((ð â ð ⧠ðž â ð) â (Vtxââšð, ðžâ©) = ð) | |
4 | 3 | eleq1d 2819 | . . 3 ⢠((ð â ð ⧠ðž â ð) â ((Vtxââšð, ðžâ©) â Fin â ð â Fin)) |
5 | 4 | anbi2d 630 | . 2 ⢠((ð â ð ⧠ðž â ð) â ((âšð, ðžâ© â USGraph ⧠(Vtxââšð, ðžâ©) â Fin) â (âšð, ðžâ© â USGraph ⧠ð â Fin))) |
6 | 2, 5 | bitrid 283 | 1 ⢠((ð â ð ⧠ðž â ð) â (âšð, ðžâ© â FinUSGraph â (âšð, ðžâ© â USGraph ⧠ð â Fin))) |
Colors of variables: wff setvar class |
Syntax hints: â wi 4 â wb 205 ⧠wa 397 â wcel 2107 âšcop 4635 âcfv 6544 Fincfn 8939 Vtxcvtx 28256 USGraphcusgr 28409 FinUSGraphcfusgr 28573 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 ax-un 7725 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-iota 6496 df-fun 6546 df-fv 6552 df-1st 7975 df-vtx 28258 df-fusgr 28574 |
This theorem is referenced by: fusgrfis 28587 |
Copyright terms: Public domain | W3C validator |