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Mirrors > Home > MPE Home > Th. List > isuspgr | Structured version Visualization version Unicode version |
Description: The property of being a simple pseudograph. (Contributed by Alexander van der Vekens, 10-Aug-2017.) (Revised by AV, 13-Oct-2020.) |
Ref | Expression |
---|---|
isuspgr.v | Vtx |
isuspgr.e | iEdg |
Ref | Expression |
---|---|
isuspgr | USPGraph |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-uspgr 26045 | . . 3 USPGraph Vtx iEdg | |
2 | 1 | eleq2i 2693 | . 2 USPGraph Vtx iEdg |
3 | fveq2 6191 | . . . . 5 iEdg iEdg | |
4 | isuspgr.e | . . . . 5 iEdg | |
5 | 3, 4 | syl6eqr 2674 | . . . 4 iEdg |
6 | 3 | dmeqd 5326 | . . . . 5 iEdg iEdg |
7 | 4 | eqcomi 2631 | . . . . . 6 iEdg |
8 | 7 | dmeqi 5325 | . . . . 5 iEdg |
9 | 6, 8 | syl6eq 2672 | . . . 4 iEdg |
10 | fveq2 6191 | . . . . . . . 8 Vtx Vtx | |
11 | isuspgr.v | . . . . . . . 8 Vtx | |
12 | 10, 11 | syl6eqr 2674 | . . . . . . 7 Vtx |
13 | 12 | pweqd 4163 | . . . . . 6 Vtx |
14 | 13 | difeq1d 3727 | . . . . 5 Vtx |
15 | 14 | rabeqdv 3194 | . . . 4 Vtx |
16 | 5, 9, 15 | f1eq123d 6131 | . . 3 iEdg iEdg Vtx |
17 | fvexd 6203 | . . . . 5 Vtx | |
18 | fveq2 6191 | . . . . 5 Vtx Vtx | |
19 | fvexd 6203 | . . . . . 6 Vtx iEdg | |
20 | fveq2 6191 | . . . . . . 7 iEdg iEdg | |
21 | 20 | adantr 481 | . . . . . 6 Vtx iEdg iEdg |
22 | simpr 477 | . . . . . . 7 Vtx iEdg iEdg | |
23 | 22 | dmeqd 5326 | . . . . . . 7 Vtx iEdg iEdg |
24 | pweq 4161 | . . . . . . . . . 10 Vtx Vtx | |
25 | 24 | ad2antlr 763 | . . . . . . . . 9 Vtx iEdg Vtx |
26 | 25 | difeq1d 3727 | . . . . . . . 8 Vtx iEdg Vtx |
27 | 26 | rabeqdv 3194 | . . . . . . 7 Vtx iEdg Vtx |
28 | 22, 23, 27 | f1eq123d 6131 | . . . . . 6 Vtx iEdg iEdg iEdg Vtx |
29 | 19, 21, 28 | sbcied2 3473 | . . . . 5 Vtx iEdg iEdg iEdg Vtx |
30 | 17, 18, 29 | sbcied2 3473 | . . . 4 Vtx iEdg iEdg iEdg Vtx |
31 | 30 | cbvabv 2747 | . . 3 Vtx iEdg iEdg iEdg Vtx |
32 | 16, 31 | elab2g 3353 | . 2 Vtx iEdg |
33 | 2, 32 | syl5bb 272 | 1 USPGraph |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wb 196 wa 384 wceq 1483 wcel 1990 cab 2608 crab 2916 cvv 3200 wsbc 3435 cdif 3571 c0 3915 cpw 4158 csn 4177 class class class wbr 4653 cdm 5114 wf1 5885 cfv 5888 cle 10075 c2 11070 chash 13117 Vtxcvtx 25874 iEdgciedg 25875 USPGraph cuspgr 26043 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-nul 4789 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fv 5896 df-uspgr 26045 |
This theorem is referenced by: uspgrf 26049 isuspgrop 26056 uspgrushgr 26070 uspgrupgr 26071 uspgrupgrushgr 26072 usgruspgr 26073 uspgr1e 26136 |
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