Theorem gropd 15971

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 4326	. . 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 15946	. . . 4 |- ( ( V e. U /\ E e. W ) -> (Vtx ` <. V , E >. ) = V )
7		opiedgfv 15949	. . . 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 2375	. . 3 |- F/_ g <. V , E >.
11		nfv 1577	. . . 4 |- F/ g ( (Vtx ` <. V , E >. ) = V /\ (iEdg ` <. V , E >. ) = E )
12		nfsbc1v 3051	. . . 4 |- F/ g [. <. V , E >. / g ]. ps
13	11, 12	nfim 1621	. . 3 |- F/ g ( ( (Vtx ` <. V , E >. ) = V /\ (iEdg ` <. V , E >. ) = E ) -> [. <. V , E >. / g ]. ps )
14		fveqeq2 5657	. . . . 5 |- ( g = <. V , E >. -> ( (Vtx ` g ) = V <-> (Vtx ` <. V , E >. ) = V ) )
15		fveqeq2 5657	. . . . 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 3042	. . . 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 2889	. 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 1396   = wceq 1398   e. wcel 2202   _Vcvv 2803   [.wsbc 3032   <.cop 3676   ` cfv 5333  Vtxcvtx 15936  iEdgciedg 15937

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-cnex 8166  ax-resscn 8167  ax-1cn 8168  ax-1re 8169  ax-icn 8170  ax-addcl 8171  ax-addrcl 8172  ax-mulcl 8173  ax-addcom 8175  ax-mulcom 8176  ax-addass 8177  ax-mulass 8178  ax-distr 8179  ax-i2m1 8180  ax-1rid 8182  ax-0id 8183  ax-rnegex 8184  ax-cnre 8186

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-if 3608  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-fo 5339  df-fv 5341  df-riota 5981  df-ov 6031  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-sub 8394  df-inn 9186  df-2 9244  df-3 9245  df-4 9246  df-5 9247  df-6 9248  df-7 9249  df-8 9250  df-9 9251  df-n0 9445  df-dec 9656  df-ndx 13148  df-slot 13149  df-base 13151  df-edgf 15929  df-vtx 15938  df-iedg 15939

This theorem is referenced by:  gropeld  15973