Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > upgrspanop | Structured version Visualization version Unicode version |
Description: A spanning subgraph of a pseudograph represented by an ordered pair is a pseudograph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by AV, 13-Oct-2020.) |
Ref | Expression |
---|---|
uhgrspanop.v | Vtx |
uhgrspanop.e | iEdg |
Ref | Expression |
---|---|
upgrspanop | UPGraph UPGraph |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uhgrspanop.v | . . . . 5 Vtx | |
2 | uhgrspanop.e | . . . . 5 iEdg | |
3 | vex 3203 | . . . . . 6 | |
4 | 3 | a1i 11 | . . . . 5 UPGraph Vtx iEdg |
5 | simprl 794 | . . . . 5 UPGraph Vtx iEdg Vtx | |
6 | simprr 796 | . . . . 5 UPGraph Vtx iEdg iEdg | |
7 | simpl 473 | . . . . 5 UPGraph Vtx iEdg UPGraph | |
8 | 1, 2, 4, 5, 6, 7 | upgrspan 26185 | . . . 4 UPGraph Vtx iEdg UPGraph |
9 | 8 | ex 450 | . . 3 UPGraph Vtx iEdg UPGraph |
10 | 9 | alrimiv 1855 | . 2 UPGraph Vtx iEdg UPGraph |
11 | fvex 6201 | . . . 4 Vtx | |
12 | 1, 11 | eqeltri 2697 | . . 3 |
13 | 12 | a1i 11 | . 2 UPGraph |
14 | fvex 6201 | . . . . 5 iEdg | |
15 | 2, 14 | eqeltri 2697 | . . . 4 |
16 | 15 | resex 5443 | . . 3 |
17 | 16 | a1i 11 | . 2 UPGraph |
18 | 10, 13, 17 | gropeld 25925 | 1 UPGraph UPGraph |
Colors of variables: wff setvar class |
Syntax hints: wi 4 wa 384 wceq 1483 wcel 1990 cvv 3200 cop 4183 cres 5116 cfv 5888 Vtxcvtx 25874 iEdgciedg 25875 UPGraph cupgr 25975 |
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-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
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-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 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-mpt 4730 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-fv 5896 df-1st 7168 df-2nd 7169 df-vtx 25876 df-iedg 25877 df-edg 25940 df-uhgr 25953 df-upgr 25977 df-subgr 26160 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |