Theorem grstructd2dom 15905

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 15886	. . . . 5 |- ( ( S e. X /\ Fun ( S \ { (/) } ) /\ 2o ~<_ dom S ) -> (Vtx ` S ) = ( Base ` S ) )
6	1, 3, 4, 5	syl3anc 1273	. . . 4 |- ( ph -> (Vtx ` S ) = ( Base ` S ) )
7		grstructd.b	. . . 4 |- ( ph -> ( Base ` S ) = V )
8	6, 7	eqtrd 2264	. . 3 |- ( ph -> (Vtx ` S ) = V )
9		funiedgdm2domval 15887	. . . . 5 |- ( ( S e. X /\ Fun ( S \ { (/) } ) /\ 2o ~<_ dom S ) -> (iEdg ` S ) = (.ef ` S ) )
10	1, 3, 4, 9	syl3anc 1273	. . . 4 |- ( ph -> (iEdg ` S ) = (.ef ` S ) )
11		grstructd.e	. . . 4 |- ( ph -> (.ef ` S ) = E )
12	10, 11	eqtrd 2264	. . 3 |- ( ph -> (iEdg ` S ) = E )
13	8, 12	jca 306	. 2 |- ( ph -> ( (Vtx ` S ) = V /\ (iEdg ` S ) = E ) )
14		nfcv 2374	. . 3 |- F/_ g S
15		nfv 1576	. . . 4 |- F/ g ( (Vtx ` S ) = V /\ (iEdg ` S ) = E )
16		nfsbc1v 3050	. . . 4 |- F/ g [. S / g ]. ps
17	15, 16	nfim 1620	. . 3 |- F/ g ( ( (Vtx ` S ) = V /\ (iEdg ` S ) = E ) -> [. S / g ]. ps )
18		fveqeq2 5648	. . . . 5 |- ( g = S -> ( (Vtx ` g ) = V <-> (Vtx ` S ) = V ) )
19		fveqeq2 5648	. . . . 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 3041	. . . 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 2888	. 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 1395   = wceq 1397   e. wcel 2202   [.wsbc 3031   \ cdif 3197   (/)c0 3494   {csn 3669   class class class wbr 4088   dom cdm 4725   Fun wfun 5320   ` cfv 5326   2oc2o 6576   ~<_ cdom 6908   Basecbs 13087  .efcedgf 15861  Vtxcvtx 15869  iEdgciedg 15870

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-cnex 8123  ax-resscn 8124  ax-1cn 8125  ax-1re 8126  ax-icn 8127  ax-addcl 8128  ax-addrcl 8129  ax-mulcl 8130  ax-addcom 8132  ax-mulcom 8133  ax-addass 8134  ax-mulass 8135  ax-distr 8136  ax-i2m1 8137  ax-1rid 8139  ax-0id 8140  ax-rnegex 8141  ax-cnre 8143

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-csb 3128  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-if 3606  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-mpt 4152  df-id 4390  df-suc 4468  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-fv 5334  df-riota 5971  df-ov 6021  df-oprab 6022  df-mpo 6023  df-1st 6303  df-2nd 6304  df-1o 6582  df-2o 6583  df-dom 6911  df-sub 8352  df-inn 9144  df-2 9202  df-3 9203  df-4 9204  df-5 9205  df-6 9206  df-7 9207  df-8 9208  df-9 9209  df-n0 9403  df-dec 9612  df-ndx 13090  df-slot 13091  df-base 13093  df-edgf 15862  df-vtx 15871  df-iedg 15872

This theorem is referenced by:  grstructeld2dom  15907