Theorem gropeld 39288

Description: If any representation of a graph with vertices V and edges E is an element of an arbitrary class C, then the ordered pair <. V , E >. of the set of vertices and the set of edges (which is such a representation of a graph with vertices V and edges E) is an element of this class C. (Contributed by AV, 11-Oct-2020.)

Hypotheses

Ref	Expression
gropeld.g	|- ( ph -> A. g ( ( (Vtx ` g ) = V /\ (iEdg ` g ) = E ) -> g e. C ) )
gropeld.v	|- ( ph -> V e. U )
gropeld.e	|- ( ph -> E e. W )

Assertion

Ref	Expression
gropeld	|- ( ph -> <. V , E >. e. C )

Distinct variable groups:   C, g   g, E   g, V   ph, g

Allowed substitution hints:   U( g)   W( g)

Proof of Theorem gropeld

Step	Hyp	Ref	Expression
1		gropeld.g	. . 3 |- ( ph -> A. g ( ( (Vtx ` g ) = V /\ (iEdg ` g ) = E ) -> g e. C ) )
2		gropeld.v	. . 3 |- ( ph -> V e. U )
3		gropeld.e	. . 3 |- ( ph -> E e. W )
4	1, 2, 3	gropd 39286	. 2 |- ( ph -> [. <. V , E >. / g ]. g e. C )
5		sbcel1v 3314	. 2 |- ( [. <. V , E >. / g ]. g e. C <-> <. V , E >. e. C )
6	4, 5	sylib 201	1 |- ( ph -> <. V , E >. e. C )

Syntax hints:   -> wi 4   /\ wa 376   A.wal 1450   = wceq 1452   e. 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