Theorem grstructd2dom 15864

Description: If any representation of a graph with vertices V and edges E has a certain property ps, then any structure with base set V and value E in the slot for edge functions (which is such a representation of a graph with vertices V and edges E) has this property. (Contributed by AV, 12-Oct-2020.) (Revised by AV, 9-Jun-2021.)

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 )
grstructd.s	|- ( ph -> S e. X )
grstructd.f	|- ( ph -> Fun ( S \ { (/) } ) )
grstructd2dom.d	|- ( ph -> 2o ~<_ dom S )
grstructd.b	|- ( ph -> ( Base ` S ) = V )
grstructd.e	|- ( ph -> (.ef ` S ) = E )

Assertion

Ref	Expression
grstructd2dom	|- ( ph -> [. S / g ]. ps )

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

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

Proof of Theorem grstructd2dom

Step	Hyp	Ref	Expression
1		grstructd.s	. 2 |- ( ph -> S e. X )
2		gropd.g	. 2 |- ( ph -> A. g ( ( (Vtx ` g ) = V /\ (iEdg ` g ) = E ) -> ps ) )
3		grstructd.f	. . . . 5 |- ( ph -> Fun ( S \ { (/) } ) )
4		grstructd2dom.d	. . . . 5 |- ( ph -> 2o ~<_ dom S )
5		funvtxdm2domval 15845	. . . . 5 |- ( ( S e. X /\ Fun ( S \ { (/) } ) /\ 2o ~<_ dom S ) -> (Vtx ` S ) = ( Base ` S ) )
6	1, 3, 4, 5	syl3anc 1271	. . . 4 |- ( ph -> (Vtx ` S ) = ( Base ` S ) )
7		grstructd.b	. . . 4 |- ( ph -> ( Base ` S ) = V )
8	6, 7	eqtrd 2262	. . 3 |- ( ph -> (Vtx ` S ) = V )
9		funiedgdm2domval 15846	. . . . 5 |- ( ( S e. X /\ Fun ( S \ { (/) } ) /\ 2o ~<_ dom S ) -> (iEdg ` S ) = (.ef ` S ) )
10	1, 3, 4, 9	syl3anc 1271	. . . 4 |- ( ph -> (iEdg ` S ) = (.ef ` S ) )
11		grstructd.e	. . . 4 |- ( ph -> (.ef ` S ) = E )
12	10, 11	eqtrd 2262	. . 3 |- ( ph -> (iEdg ` S ) = E )
13	8, 12	jca 306	. 2 |- ( ph -> ( (Vtx ` S ) = V /\ (iEdg ` S ) = E ) )
14		nfcv 2372	. . 3 |- F/_ g S
15		nfv 1574	. . . 4 |- F/ g ( (Vtx ` S ) = V /\ (iEdg ` S ) = E )
16		nfsbc1v 3047	. . . 4 |- F/ g [. S / g ]. ps
17	15, 16	nfim 1618	. . 3 |- F/ g ( ( (Vtx ` S ) = V /\ (iEdg ` S ) = E ) -> [. S / g ]. ps )
18		fveqeq2 5638	. . . . 5 |- ( g = S -> ( (Vtx ` g ) = V <-> (Vtx ` S ) = V ) )
19		fveqeq2 5638	. . . . 5 |- ( g = S -> ( (iEdg ` g ) = E <-> (iEdg ` S ) = E ) )
20	18, 19	anbi12d 473	. . . 4 |- ( g = S -> ( ( (Vtx ` g ) = V /\ (iEdg ` g ) = E ) <-> ( (Vtx ` S ) = V /\ (iEdg ` S ) = E ) ) )
21		sbceq1a 3038	. . . 4 |- ( g = S -> ( ps <-> [. S / g ]. ps ) )
22	20, 21	imbi12d 234	. . 3 |- ( g = S -> ( ( ( (Vtx ` g ) = V /\ (iEdg ` g ) = E ) -> ps ) <-> ( ( (Vtx ` S ) = V /\ (iEdg ` S ) = E ) -> [. S / g ]. ps ) ) )
23	14, 17, 22	spcgf 2885	. 2 |- ( S e. X -> ( A. g ( ( (Vtx ` g ) = V /\ (iEdg ` g ) = E ) -> ps ) -> ( ( (Vtx ` S ) = V /\ (iEdg ` S ) = E ) -> [. S / g ]. ps ) ) )
24	1, 2, 13, 23	syl3c 63	1 |- ( ph -> [. S / g ]. ps )

Syntax hints:   -> wi 4   /\ wa 104   A.wal 1393   = wceq 1395   e. wcel 2200   [.wsbc 3028   \ cdif 3194   (/)c0 3491   {csn 3666   class class class wbr 4083   dom cdm 4719   Fun wfun 5312   ` cfv 5318   2oc2o 6562   ~<_ cdom 6894   Basecbs 13047  .efcedgf 15820  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-nul 4210  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-nul 3492  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-suc 4462  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-f1 5323  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-1o 6568  df-2o 6569  df-dom 6897  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:  grstructeld2dom  15866