Theorem gropd 39286

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 4664	. . 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 39259	. . . 4 |- ( ( V e. U /\ E e. W ) -> (Vtx ` <. V , E >. ) = V )
7		opiedgfv 39262	. . . 4 |- ( ( V e. U /\ E e. W ) -> (iEdg ` <. V , E >. ) = E )
8	6, 7	jca 541	. . 3 |- ( ( V e. U /\ E e. W ) -> ( (Vtx ` <. V , E >. ) = V /\ (iEdg ` <. V , E >. ) = E ) )
9	4, 5, 8	syl2anc 673	. 2 |- ( ph -> ( (Vtx ` <. V , E >. ) = V /\ (iEdg ` <. V , E >. ) = E ) )
10		nfcv 2612	. . 3 |- F/_ g <. V , E >.
11		nfv 1769	. . . 4 |- F/ g ( (Vtx ` <. V , E >. ) = V /\ (iEdg ` <. V , E >. ) = E )
12		nfsbc1v 3275	. . . 4 |- F/ g [. <. V , E >. / g ]. ps
13	11, 12	nfim 2023	. . 3 |- F/ g ( ( (Vtx ` <. V , E >. ) = V /\ (iEdg ` <. V , E >. ) = E ) -> [. <. V , E >. / g ]. ps )
14		fveq2 5879	. . . . . 6 |- ( g = <. V , E >. -> (Vtx ` g ) = (Vtx ` <. V , E >. ) )
15	14	eqeq1d 2473	. . . . 5 |- ( g = <. V , E >. -> ( (Vtx ` g ) = V <-> (Vtx ` <. V , E >. ) = V ) )
16		fveq2 5879	. . . . . 6 |- ( g = <. V , E >. -> (iEdg ` g ) = (iEdg ` <. V , E >. ) )
17	16	eqeq1d 2473	. . . . 5 |- ( g = <. V , E >. -> ( (iEdg ` g ) = E <-> (iEdg ` <. V , E >. ) = E ) )
18	15, 17	anbi12d 725	. . . 4 |- ( g = <. V , E >. -> ( ( (Vtx ` g ) = V /\ (iEdg ` g ) = E ) <-> ( (Vtx ` <. V , E >. ) = V /\ (iEdg ` <. V , E >. ) = E ) ) )
19		sbceq1a 3266	. . . 4 |- ( g = <. V , E >. -> ( ps <-> [. <. V , E >. / g ]. ps ) )
20	18, 19	imbi12d 327	. . 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 3115	. 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 62	1 |- ( ph -> [. <. V , E >. / g ]. ps )

Syntax hints:   -> wi 4   /\ wa 376   A.wal 1450   = wceq 1452   e. wcel 1904   _Vcvv 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