Theorem gropd 25923

Description: If any representation of a graph with vertices V and edges E has a certain property ps, 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) has this property. (Contributed by AV, 11-Oct-2020.)

Hypotheses

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

Assertion

Ref	Expression
gropd	|- ( ph -> [. <. V , E >. / g ]. ps )

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

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

Proof of Theorem gropd

Step	Hyp	Ref	Expression
1		opex 4932	. . 3 |- <. V , E >. e. _V
2	1	a1i 11	. 2 |- ( ph -> <. V , E >. e. _V )
3		gropd.g	. 2 |- ( ph -> A. g ( ( (Vtx ` g ) = V /\ (iEdg ` g ) = E ) -> ps ) )
4		gropd.v	. . 3 |- ( ph -> V e. U )
5		gropd.e	. . 3 |- ( ph -> E e. W )
6		opvtxfv 25884	. . . 4 |- ( ( V e. U /\ E e. W ) -> (Vtx ` <. V , E >. ) = V )
7		opiedgfv 25887	. . . 4 |- ( ( V e. U /\ E e. W ) -> (iEdg ` <. V , E >. ) = E )
8	6, 7	jca 554	. . 3 |- ( ( V e. U /\ E e. W ) -> ( (Vtx ` <. V , E >. ) = V /\ (iEdg ` <. V , E >. ) = E ) )
9	4, 5, 8	syl2anc 693	. 2 |- ( ph -> ( (Vtx ` <. V , E >. ) = V /\ (iEdg ` <. V , E >. ) = E ) )
10		nfcv 2764	. . 3 |- F/_ g <. V , E >.
11		nfv 1843	. . . 4 |- F/ g ( (Vtx ` <. V , E >. ) = V /\ (iEdg ` <. V , E >. ) = E )
12		nfsbc1v 3455	. . . 4 |- F/ g [. <. V , E >. / g ]. ps
13	11, 12	nfim 1825	. . 3 |- F/ g ( ( (Vtx ` <. V , E >. ) = V /\ (iEdg ` <. V , E >. ) = E ) -> [. <. V , E >. / g ]. ps )
14		fveq2 6191	. . . . . 6 |- ( g = <. V , E >. -> (Vtx ` g ) = (Vtx ` <. V , E >. ) )
15	14	eqeq1d 2624	. . . . 5 |- ( g = <. V , E >. -> ( (Vtx ` g ) = V <-> (Vtx ` <. V , E >. ) = V ) )
16		fveq2 6191	. . . . . 6 |- ( g = <. V , E >. -> (iEdg ` g ) = (iEdg ` <. V , E >. ) )
17	16	eqeq1d 2624	. . . . 5 |- ( g = <. V , E >. -> ( (iEdg ` g ) = E <-> (iEdg ` <. V , E >. ) = E ) )
18	15, 17	anbi12d 747	. . . 4 |- ( g = <. V , E >. -> ( ( (Vtx ` g ) = V /\ (iEdg ` g ) = E ) <-> ( (Vtx ` <. V , E >. ) = V /\ (iEdg ` <. V , E >. ) = E ) ) )
19		sbceq1a 3446	. . . 4 |- ( g = <. V , E >. -> ( ps <-> [. <. V , E >. / g ]. ps ) )
20	18, 19	imbi12d 334	. . 3 |- ( g = <. V , E >. -> ( ( ( (Vtx ` g ) = V /\ (iEdg ` g ) = E ) -> ps ) <-> ( ( (Vtx ` <. V , E >. ) = V /\ (iEdg ` <. V , E >. ) = E ) -> [. <. V , E >. / g ]. ps ) ) )
21	10, 13, 20	spcgf 3288	. 2 |- ( <. V , E >. e. _V -> ( A. g ( ( (Vtx ` g ) = V /\ (iEdg ` g ) = E ) -> ps ) -> ( ( (Vtx ` <. V , E >. ) = V /\ (iEdg ` <. V , E >. ) = E ) -> [. <. V , E >. / g ]. ps ) ) )
22	2, 3, 9, 21	syl3c 66	1 |- ( ph -> [. <. V , E >. / g ]. ps )

Syntax hints:   -> wi 4   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   _Vcvv 3200   [.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:  gropeld  25925