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Theorem gropd 39286
Description: If any representation of a graph with vertices  V and edges  E has a certain property  ps, then the ordered pair  <. V ,  E >. of the set of vertices and the set of edges (which is such a representation of a graph with vertices  V and edges  E) has this property. (Contributed by AV, 11-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 )
Assertion
Ref Expression
gropd  |-  ( ph  ->  [. <. V ,  E >.  /  g ]. ps )
Distinct variable groups:    g, E    g, V    ph, g
Allowed substitution hints:    ps( g)    U( g)    W( g)

Proof of Theorem gropd
StepHypRef Expression
1 opex 4664 . . 3  |-  <. V ,  E >.  e.  _V
21a1i 11 . 2  |-  ( ph  -> 
<. V ,  E >.  e. 
_V )
3 gropd.g . 2  |-  ( ph  ->  A. g ( ( (Vtx `  g )  =  V  /\  (iEdg `  g )  =  E )  ->  ps )
)
4 gropd.v . . 3  |-  ( ph  ->  V  e.  U )
5 gropd.e . . 3  |-  ( ph  ->  E  e.  W )
6 opvtxfv 39259 . . . 4  |-  ( ( V  e.  U  /\  E  e.  W )  ->  (Vtx `  <. V ,  E >. )  =  V )
7 opiedgfv 39262 . . . 4  |-  ( ( V  e.  U  /\  E  e.  W )  ->  (iEdg `  <. V ,  E >. )  =  E )
86, 7jca 541 . . 3  |-  ( ( V  e.  U  /\  E  e.  W )  ->  ( (Vtx `  <. V ,  E >. )  =  V  /\  (iEdg ` 
<. V ,  E >. )  =  E ) )
94, 5, 8syl2anc 673 . 2  |-  ( ph  ->  ( (Vtx `  <. V ,  E >. )  =  V  /\  (iEdg ` 
<. V ,  E >. )  =  E ) )
10 nfcv 2612 . . 3  |-  F/_ g <. V ,  E >.
11 nfv 1769 . . . 4  |-  F/ g ( (Vtx `  <. V ,  E >. )  =  V  /\  (iEdg ` 
<. V ,  E >. )  =  E )
12 nfsbc1v 3275 . . . 4  |-  F/ g
[. <. V ,  E >.  /  g ]. ps
1311, 12nfim 2023 . . 3  |-  F/ g ( ( (Vtx `  <. V ,  E >. )  =  V  /\  (iEdg ` 
<. V ,  E >. )  =  E )  ->  [. <. V ,  E >.  /  g ]. ps )
14 fveq2 5879 . . . . . 6  |-  ( g  =  <. V ,  E >.  ->  (Vtx `  g
)  =  (Vtx `  <. V ,  E >. ) )
1514eqeq1d 2473 . . . . 5  |-  ( g  =  <. V ,  E >.  ->  ( (Vtx `  g )  =  V  <-> 
(Vtx `  <. V ,  E >. )  =  V ) )
16 fveq2 5879 . . . . . 6  |-  ( g  =  <. V ,  E >.  ->  (iEdg `  g
)  =  (iEdg `  <. V ,  E >. ) )
1716eqeq1d 2473 . . . . 5  |-  ( g  =  <. V ,  E >.  ->  ( (iEdg `  g )  =  E  <-> 
(iEdg `  <. V ,  E >. )  =  E ) )
1815, 17anbi12d 725 . . . 4  |-  ( g  =  <. V ,  E >.  ->  ( ( (Vtx
`  g )  =  V  /\  (iEdg `  g )  =  E )  <->  ( (Vtx `  <. V ,  E >. )  =  V  /\  (iEdg ` 
<. V ,  E >. )  =  E ) ) )
19 sbceq1a 3266 . . . 4  |-  ( g  =  <. V ,  E >.  ->  ( ps  <->  [. <. V ,  E >.  /  g ]. ps ) )
2018, 19imbi12d 327 . . 3  |-  ( g  =  <. V ,  E >.  ->  ( ( ( (Vtx `  g )  =  V  /\  (iEdg `  g )  =  E )  ->  ps )  <->  ( ( (Vtx `  <. V ,  E >. )  =  V  /\  (iEdg ` 
<. V ,  E >. )  =  E )  ->  [. <. V ,  E >.  /  g ]. ps ) ) )
2110, 13, 20spcgf 3115 . 2  |-  ( <. V ,  E >.  e. 
_V  ->  ( A. g
( ( (Vtx `  g )  =  V  /\  (iEdg `  g
)  =  E )  ->  ps )  -> 
( ( (Vtx `  <. V ,  E >. )  =  V  /\  (iEdg ` 
<. V ,  E >. )  =  E )  ->  [. <. V ,  E >.  /  g ]. ps ) ) )
222, 3, 9, 21syl3c 62 1  |-  ( ph  ->  [. <. V ,  E >.  /  g ]. ps )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376   A.wal 1450    = wceq 1452    e. wcel 1904   _Vcvv 3031   [.wsbc 3255   <.cop 3965   ` cfv 5589  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
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  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-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-iota 5553  df-fun 5591  df-fv 5597  df-1st 6812  df-2nd 6813  df-vtx 39253  df-iedg 39254
This theorem is referenced by:  gropeld  39288
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