Theorem gropeld 25925

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 25923	. 2 |- ( ph -> [. <. V , E >. / g ]. g e. C )
5		sbcel1v 3495	. 2 |- ( [. <. V , E >. / g ]. g e. C <-> <. V , E >. e. C )
6	4, 5	sylib 208	1 |- ( ph -> <. V , E >. e. C )

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   [.wsbc 3435   <.cop 4183   ` cfv 5888  Vtxcvtx 25874  iEdgciedg 25875

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-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-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-iota 5851  df-fun 5890  df-fv 5896  df-1st 7168  df-2nd 7169  df-vtx 25876  df-iedg 25877

This theorem is referenced by:  upgr0eopALT  26011  upgr1eopALT  26012  upgrspanop  26189  umgrspanop  26190  usgrspanop  26191  cplgrop  26333