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Theorem gropeld 39288
 Description: If any representation of a graph with vertices and edges is an element of an arbitrary class , 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 ) is an element of this class . (Contributed by AV, 11-Oct-2020.)
Hypotheses
Ref Expression
gropeld.g Vtx iEdg
gropeld.v
gropeld.e
Assertion
Ref Expression
gropeld
Distinct variable groups:   ,   ,   ,   ,
Allowed substitution hints:   ()   ()

Proof of Theorem gropeld
StepHypRef Expression
1 gropeld.g . . 3 Vtx iEdg
2 gropeld.v . . 3
3 gropeld.e . . 3
41, 2, 3gropd 39286 . 2
5 sbcel1v 3314 . 2
64, 5sylib 201 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 376  wal 1450   wceq 1452   wcel 1904  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:  upgr0eopALT  39361  upgr1eopALT  39362  upgrunop  39364  umgrunop  39366  upgrspanop  39533  umgrspanop  39534  usgrspanop  39535  cplgrop  39669
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