Theorem gropd 15863

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 4314	. . 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 15838	. . . 4 |- ( ( V e. U /\ E e. W ) -> (Vtx ` <. V , E >. ) = V )
7		opiedgfv 15841	. . . 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 2372	. . 3 |- F/_ g <. V , E >.
11		nfv 1574	. . . 4 |- F/ g ( (Vtx ` <. V , E >. ) = V /\ (iEdg ` <. V , E >. ) = E )
12		nfsbc1v 3047	. . . 4 |- F/ g [. <. V , E >. / g ]. ps
13	11, 12	nfim 1618	. . 3 |- F/ g ( ( (Vtx ` <. V , E >. ) = V /\ (iEdg ` <. V , E >. ) = E ) -> [. <. V , E >. / g ]. ps )
14		fveqeq2 5638	. . . . 5 |- ( g = <. V , E >. -> ( (Vtx ` g ) = V <-> (Vtx ` <. V , E >. ) = V ) )
15		fveqeq2 5638	. . . . 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 3038	. . . 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 2885	. 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 1393   = wceq 1395   e. wcel 2200   _Vcvv 2799   [.wsbc 3028   <.cop 3669   ` cfv 5318  Vtxcvtx 15828  iEdgciedg 15829

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8101  ax-resscn 8102  ax-1cn 8103  ax-1re 8104  ax-icn 8105  ax-addcl 8106  ax-addrcl 8107  ax-mulcl 8108  ax-addcom 8110  ax-mulcom 8111  ax-addass 8112  ax-mulass 8113  ax-distr 8114  ax-i2m1 8115  ax-1rid 8117  ax-0id 8118  ax-rnegex 8119  ax-cnre 8121

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-mpt 4147  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-fo 5324  df-fv 5326  df-riota 5960  df-ov 6010  df-oprab 6011  df-mpo 6012  df-1st 6292  df-2nd 6293  df-sub 8330  df-inn 9122  df-2 9180  df-3 9181  df-4 9182  df-5 9183  df-6 9184  df-7 9185  df-8 9186  df-9 9187  df-n0 9381  df-dec 9590  df-ndx 13050  df-slot 13051  df-base 13053  df-edgf 15821  df-vtx 15830  df-iedg 15831

This theorem is referenced by:  gropeld  15865