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Theorem gropd 39286
 Description: If any representation of a graph with vertices and edges has a certain property , then the ordered pair of the set of vertices and the set of edges (which is such a representation of a graph with vertices and edges ) has this property. (Contributed by AV, 11-Oct-2020.)
Hypotheses
Ref Expression
gropd.g Vtx iEdg
gropd.v
gropd.e
Assertion
Ref Expression
gropd
Distinct variable groups:   ,   ,   ,
Allowed substitution hints:   ()   ()   ()

Proof of Theorem gropd
StepHypRef Expression
1 opex 4664 . . 3
21a1i 11 . 2
3 gropd.g . 2 Vtx iEdg
4 gropd.v . . 3
5 gropd.e . . 3
6 opvtxfv 39259 . . . 4 Vtx
7 opiedgfv 39262 . . . 4 iEdg
86, 7jca 541 . . 3 Vtx iEdg
94, 5, 8syl2anc 673 . 2 Vtx iEdg
10 nfcv 2612 . . 3
11 nfv 1769 . . . 4 Vtx iEdg
12 nfsbc1v 3275 . . . 4
1311, 12nfim 2023 . . 3 Vtx iEdg
14 fveq2 5879 . . . . . 6 Vtx Vtx
1514eqeq1d 2473 . . . . 5 Vtx Vtx
16 fveq2 5879 . . . . . 6 iEdg iEdg
1716eqeq1d 2473 . . . . 5 iEdg iEdg
1815, 17anbi12d 725 . . . 4 Vtx iEdg Vtx iEdg
19 sbceq1a 3266 . . . 4
2018, 19imbi12d 327 . . 3 Vtx iEdg Vtx iEdg
2110, 13, 20spcgf 3115 . 2 Vtx iEdg Vtx iEdg
222, 3, 9, 21syl3c 62 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 376  wal 1450   wceq 1452   wcel 1904  cvv 3031  wsbc 3255  cop 3965  cfv 5589  Vtxcvtx 39251  iEdgciedg 39252 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-iota 5553  df-fun 5591  df-fv 5597  df-1st 6812  df-2nd 6813  df-vtx 39253  df-iedg 39254 This theorem is referenced by:  gropeld  39288
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