Theorem gropd 15586

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		gropd.v	. . 3 |- ( ph -> V e. U )
2		gropd.e	. . 3 |- ( ph -> E e. W )
3		opexg 4271	. . 3 |- ( ( V e. U /\ E e. W ) -> <. V , E >. e. _V )
4	1, 2, 3	syl2anc 411	. 2 |- ( ph -> <. V , E >. e. _V )
5		gropd.g	. 2 |- ( ph -> A. g ( ( (Vtx ` g ) = V /\ (iEdg ` g ) = E ) -> ps ) )
6		opvtxfv 15561	. . . 4 |- ( ( V e. U /\ E e. W ) -> (Vtx ` <. V , E >. ) = V )
7		opiedgfv 15564	. . . 4 |- ( ( V e. U /\ E e. W ) -> (iEdg ` <. V , E >. ) = E )
8	6, 7	jca 306	. . 3 |- ( ( V e. U /\ E e. W ) -> ( (Vtx ` <. V , E >. ) = V /\ (iEdg ` <. V , E >. ) = E ) )
9	1, 2, 8	syl2anc 411	. 2 |- ( ph -> ( (Vtx ` <. V , E >. ) = V /\ (iEdg ` <. V , E >. ) = E ) )
10		nfcv 2347	. . 3 |- F/_ g <. V , E >.
11		nfv 1550	. . . 4 |- F/ g ( (Vtx ` <. V , E >. ) = V /\ (iEdg ` <. V , E >. ) = E )
12		nfsbc1v 3016	. . . 4 |- F/ g [. <. V , E >. / g ]. ps
13	11, 12	nfim 1594	. . 3 |- F/ g ( ( (Vtx ` <. V , E >. ) = V /\ (iEdg ` <. V , E >. ) = E ) -> [. <. V , E >. / g ]. ps )
14		fveqeq2 5584	. . . . 5 |- ( g = <. V , E >. -> ( (Vtx ` g ) = V <-> (Vtx ` <. V , E >. ) = V ) )
15		fveqeq2 5584	. . . . 5 |- ( g = <. V , E >. -> ( (iEdg ` g ) = E <-> (iEdg ` <. V , E >. ) = E ) )
16	14, 15	anbi12d 473	. . . 4 |- ( g = <. V , E >. -> ( ( (Vtx ` g ) = V /\ (iEdg ` g ) = E ) <-> ( (Vtx ` <. V , E >. ) = V /\ (iEdg ` <. V , E >. ) = E ) ) )
17		sbceq1a 3007	. . . 4 |- ( g = <. V , E >. -> ( ps <-> [. <. V , E >. / g ]. ps ) )
18	16, 17	imbi12d 234	. . 3 |- ( g = <. V , E >. -> ( ( ( (Vtx ` g ) = V /\ (iEdg ` g ) = E ) -> ps ) <-> ( ( (Vtx ` <. V , E >. ) = V /\ (iEdg ` <. V , E >. ) = E ) -> [. <. V , E >. / g ]. ps ) ) )
19	10, 13, 18	spcgf 2854	. 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 ) ) )
20	4, 5, 9, 19	syl3c 63	1 |- ( ph -> [. <. V , E >. / g ]. ps )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1370   = wceq 1372   e. wcel 2175   _Vcvv 2771   [.wsbc 2997   <.cop 3635   ` cfv 5270  Vtxcvtx 15553  iEdgciedg 15554

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-sep 4161  ax-pow 4217  ax-pr 4252  ax-un 4479  ax-setind 4584  ax-cnex 8015  ax-resscn 8016  ax-1cn 8017  ax-1re 8018  ax-icn 8019  ax-addcl 8020  ax-addrcl 8021  ax-mulcl 8022  ax-addcom 8024  ax-mulcom 8025  ax-addass 8026  ax-mulass 8027  ax-distr 8028  ax-i2m1 8029  ax-1rid 8031  ax-0id 8032  ax-rnegex 8033  ax-cnre 8035

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-fal 1378  df-nf 1483  df-sb 1785  df-eu 2056  df-mo 2057  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ne 2376  df-ral 2488  df-rex 2489  df-reu 2490  df-rab 2492  df-v 2773  df-sbc 2998  df-csb 3093  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-if 3571  df-pw 3617  df-sn 3638  df-pr 3639  df-op 3641  df-uni 3850  df-int 3885  df-br 4044  df-opab 4105  df-mpt 4106  df-id 4339  df-xp 4680  df-rel 4681  df-cnv 4682  df-co 4683  df-dm 4684  df-rn 4685  df-res 4686  df-iota 5231  df-fun 5272  df-fn 5273  df-f 5274  df-fo 5276  df-fv 5278  df-riota 5898  df-ov 5946  df-oprab 5947  df-mpo 5948  df-1st 6225  df-2nd 6226  df-sub 8244  df-inn 9036  df-2 9094  df-3 9095  df-4 9096  df-5 9097  df-6 9098  df-7 9099  df-8 9100  df-9 9101  df-n0 9295  df-dec 9504  df-ndx 12777  df-slot 12778  df-base 12780  df-edgf 15546  df-vtx 15555  df-iedg 15556

This theorem is referenced by:  gropeld  15588