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Theorem grstructd 39287
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.)
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 )
grastructd.s  |-  ( ph  ->  S  e.  X )
grastructd.f  |-  ( ph  ->  Fun  S )
grastructd.d  |-  ( ph  ->  2  <_  ( # `  dom  S ) )
grastructd.b  |-  ( ph  ->  ( Base `  S
)  =  V )
grastructd.e  |-  ( ph  ->  (.ef `  S )  =  E )
Assertion
Ref Expression
grstructd  |-  ( 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 grstructd
StepHypRef Expression
1 grastructd.s . 2  |-  ( ph  ->  S  e.  X )
2 gropd.g . 2  |-  ( ph  ->  A. g ( ( (Vtx `  g )  =  V  /\  (iEdg `  g )  =  E )  ->  ps )
)
3 grastructd.f . . . . 5  |-  ( ph  ->  Fun  S )
4 grastructd.d . . . . 5  |-  ( ph  ->  2  <_  ( # `  dom  S ) )
5 funvtxdmge2val 39269 . . . . 5  |-  ( ( S  e.  X  /\  Fun  S  /\  2  <_ 
( # `  dom  S
) )  ->  (Vtx `  S )  =  (
Base `  S )
)
61, 3, 4, 5syl3anc 1292 . . . 4  |-  ( ph  ->  (Vtx `  S )  =  ( Base `  S
) )
7 grastructd.b . . . 4  |-  ( ph  ->  ( Base `  S
)  =  V )
86, 7eqtrd 2505 . . 3  |-  ( ph  ->  (Vtx `  S )  =  V )
9 funiedgdmge2val 39270 . . . . 5  |-  ( ( S  e.  X  /\  Fun  S  /\  2  <_ 
( # `  dom  S
) )  ->  (iEdg `  S )  =  (.ef
`  S ) )
101, 3, 4, 9syl3anc 1292 . . . 4  |-  ( ph  ->  (iEdg `  S )  =  (.ef `  S )
)
11 grastructd.e . . . 4  |-  ( ph  ->  (.ef `  S )  =  E )
1210, 11eqtrd 2505 . . 3  |-  ( ph  ->  (iEdg `  S )  =  E )
138, 12jca 541 . 2  |-  ( ph  ->  ( (Vtx `  S
)  =  V  /\  (iEdg `  S )  =  E ) )
14 nfcv 2612 . . 3  |-  F/_ g S
15 nfv 1769 . . . 4  |-  F/ g ( (Vtx `  S
)  =  V  /\  (iEdg `  S )  =  E )
16 nfsbc1v 3275 . . . 4  |-  F/ g
[. S  /  g ]. ps
1715, 16nfim 2023 . . 3  |-  F/ g ( ( (Vtx `  S )  =  V  /\  (iEdg `  S
)  =  E )  ->  [. S  /  g ]. ps )
18 fveq2 5879 . . . . . 6  |-  ( g  =  S  ->  (Vtx `  g )  =  (Vtx
`  S ) )
1918eqeq1d 2473 . . . . 5  |-  ( g  =  S  ->  (
(Vtx `  g )  =  V  <->  (Vtx `  S )  =  V ) )
20 fveq2 5879 . . . . . 6  |-  ( g  =  S  ->  (iEdg `  g )  =  (iEdg `  S ) )
2120eqeq1d 2473 . . . . 5  |-  ( g  =  S  ->  (
(iEdg `  g )  =  E  <->  (iEdg `  S )  =  E ) )
2219, 21anbi12d 725 . . . 4  |-  ( g  =  S  ->  (
( (Vtx `  g
)  =  V  /\  (iEdg `  g )  =  E )  <->  ( (Vtx `  S )  =  V  /\  (iEdg `  S
)  =  E ) ) )
23 sbceq1a 3266 . . . 4  |-  ( g  =  S  ->  ( ps 
<-> 
[. S  /  g ]. ps ) )
2422, 23imbi12d 327 . . 3  |-  ( g  =  S  ->  (
( ( (Vtx `  g )  =  V  /\  (iEdg `  g
)  =  E )  ->  ps )  <->  ( (
(Vtx `  S )  =  V  /\  (iEdg `  S )  =  E )  ->  [. S  / 
g ]. ps ) ) )
2514, 17, 24spcgf 3115 . 2  |-  ( S  e.  X  ->  ( A. g ( ( (Vtx
`  g )  =  V  /\  (iEdg `  g )  =  E )  ->  ps )  ->  ( ( (Vtx `  S )  =  V  /\  (iEdg `  S
)  =  E )  ->  [. S  /  g ]. ps ) ) )
261, 2, 13, 25syl3c 62 1  |-  ( ph  ->  [. S  /  g ]. ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376   A.wal 1450    = wceq 1452    e. wcel 1904   [.wsbc 3255   class class class wbr 4395   dom cdm 4839   Fun wfun 5583   ` cfv 5589    <_ cle 9694   2c2 10681   #chash 12553   Basecbs 15199  .efcedgf 39245  Vtxcvtx 39251  iEdgciedg 39252
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-hash 12554  df-vtx 39253  df-iedg 39254
This theorem is referenced by:  grstructeld  39289
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