Theorem grstructd2dom 16289

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 16270	. . . . 5 |- ( ( S e. X /\ Fun ( S \ { (/) } ) /\ 2o ~<_ dom S ) -> (Vtx ` S ) = ( Base ` S ) )
6	1, 3, 4, 5	syl3anc 1278	. . . 4 |- ( ph -> (Vtx ` S ) = ( Base ` S ) )
7		grstructd.b	. . . 4 |- ( ph -> ( Base ` S ) = V )
8	6, 7	eqtrd 2271	. . 3 |- ( ph -> (Vtx ` S ) = V )
9		funiedgdm2domval 16271	. . . . 5 |- ( ( S e. X /\ Fun ( S \ { (/) } ) /\ 2o ~<_ dom S ) -> (iEdg ` S ) = (.ef ` S ) )
10	1, 3, 4, 9	syl3anc 1278	. . . 4 |- ( ph -> (iEdg ` S ) = (.ef ` S ) )
11		grstructd.e	. . . 4 |- ( ph -> (.ef ` S ) = E )
12	10, 11	eqtrd 2271	. . 3 |- ( ph -> (iEdg ` S ) = E )
13	8, 12	jca 306	. 2 |- ( ph -> ( (Vtx ` S ) = V /\ (iEdg ` S ) = E ) )
14		nfcv 2392	. . 3 |- F/_ g S
15		nfv 1581	. . . 4 |- F/ g ( (Vtx ` S ) = V /\ (iEdg ` S ) = E )
16		nfsbc1v 3070	. . . 4 |- F/ g [. S / g ]. ps
17	15, 16	nfim 1625	. . 3 |- F/ g ( ( (Vtx ` S ) = V /\ (iEdg ` S ) = E ) -> [. S / g ]. ps )
18		fveqeq2 5704	. . . . 5 |- ( g = S -> ( (Vtx ` g ) = V <-> (Vtx ` S ) = V ) )
19		fveqeq2 5704	. . . . 5 |- ( g = S -> ( (iEdg ` g ) = E <-> (iEdg ` S ) = E ) )
20	18, 19	anbi12d 477	. . . 4 |- ( g = S -> ( ( (Vtx ` g ) = V /\ (iEdg ` g ) = E ) <-> ( (Vtx ` S ) = V /\ (iEdg ` S ) = E ) ) )
21		sbceq1a 3061	. . . 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 2907	. 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 )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   A.wal 1400   = wceq 1402   e. wcel 2209   [.wsbc 3051   \ cdif 3217   (/)c0 3520   {csn 3709   class class class wbr 4130   dom cdm 4774   Fun wfun 5371   ` cfv 5377   2oc2o 6681   ~<_ cdom 7021   Basecbs 13352  .efcedgf 16245  Vtxcvtx 16253  iEdgciedg 16254

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-cnex 8270  ax-resscn 8271  ax-1cn 8272  ax-1re 8273  ax-icn 8274  ax-addcl 8275  ax-addrcl 8276  ax-mulcl 8277  ax-addcom 8279  ax-mulcom 8280  ax-addass 8281  ax-mulass 8282  ax-distr 8283  ax-i2m1 8284  ax-1rid 8286  ax-0id 8287  ax-rnegex 8288  ax-cnre 8290

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-if 3639  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-suc 4516  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-fv 5385  df-riota 6038  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-1o 6687  df-2o 6688  df-dom 7024  df-sub 8499  df-inn 9305  df-2 9363  df-3 9364  df-4 9365  df-5 9366  df-6 9367  df-7 9368  df-8 9369  df-9 9370  df-n0 9564  df-dec 9778  df-ndx 13355  df-slot 13356  df-base 13358  df-edgf 16246  df-vtx 16255  df-iedg 16256

This theorem is used by:  grstructeld2dom  16291