Theorem grstructd2dom 16169

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 16150	. . . . 5 |- ( ( S e. X /\ Fun ( S \ { (/) } ) /\ 2o ~<_ dom S ) -> (Vtx ` S ) = ( Base ` S ) )
6	1, 3, 4, 5	syl3anc 1274	. . . 4 |- ( ph -> (Vtx ` S ) = ( Base ` S ) )
7		grstructd.b	. . . 4 |- ( ph -> ( Base ` S ) = V )
8	6, 7	eqtrd 2267	. . 3 |- ( ph -> (Vtx ` S ) = V )
9		funiedgdm2domval 16151	. . . . 5 |- ( ( S e. X /\ Fun ( S \ { (/) } ) /\ 2o ~<_ dom S ) -> (iEdg ` S ) = (.ef ` S ) )
10	1, 3, 4, 9	syl3anc 1274	. . . 4 |- ( ph -> (iEdg ` S ) = (.ef ` S ) )
11		grstructd.e	. . . 4 |- ( ph -> (.ef ` S ) = E )
12	10, 11	eqtrd 2267	. . 3 |- ( ph -> (iEdg ` S ) = E )
13	8, 12	jca 306	. 2 |- ( ph -> ( (Vtx ` S ) = V /\ (iEdg ` S ) = E ) )
14		nfcv 2386	. . 3 |- F/_ g S
15		nfv 1577	. . . 4 |- F/ g ( (Vtx ` S ) = V /\ (iEdg ` S ) = E )
16		nfsbc1v 3064	. . . 4 |- F/ g [. S / g ]. ps
17	15, 16	nfim 1621	. . 3 |- F/ g ( ( (Vtx ` S ) = V /\ (iEdg ` S ) = E ) -> [. S / g ]. ps )
18		fveqeq2 5684	. . . . 5 |- ( g = S -> ( (Vtx ` g ) = V <-> (Vtx ` S ) = V ) )
19		fveqeq2 5684	. . . . 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 3055	. . . 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 2901	. 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 1396   = wceq 1398   e. wcel 2205   [.wsbc 3045   \ cdif 3211   (/)c0 3512   {csn 3694   class class class wbr 4114   dom cdm 4754   Fun wfun 5351   ` cfv 5357   2oc2o 6654   ~<_ cdom 6987   Basecbs 13296  .efcedgf 16125  Vtxcvtx 16133  iEdgciedg 16134

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-suc 4497  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-1o 6660  df-2o 6661  df-dom 6990  df-sub 8462  df-inn 9255  df-2 9313  df-3 9314  df-4 9315  df-5 9316  df-6 9317  df-7 9318  df-8 9319  df-9 9320  df-n0 9514  df-dec 9728  df-ndx 13299  df-slot 13300  df-base 13302  df-edgf 16126  df-vtx 16135  df-iedg 16136

This theorem is referenced by:  grstructeld2dom  16171