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Theorem vtxduhgrun 39706
Description: The degree of a vertex in the union of two hypergraphs on the same vertex set is the sum of the degrees of the vertex in each hypergraph. (Contributed by Mario Carneiro, 12-Mar-2015.) (Revised by Alexander van der Vekens, 21-Dec-2017.) (Revised by AV, 12-Dec-2020.) (Proof shortened by AV, 19-Feb-2021.)
Hypotheses
Ref Expression
vtxduhgrun.i  |-  I  =  (iEdg `  G )
vtxduhgrun.j  |-  J  =  (iEdg `  H )
vtxduhgrun.vg  |-  V  =  (Vtx `  G )
vtxduhgrun.vh  |-  ( ph  ->  (Vtx `  H )  =  V )
vtxduhgrun.vu  |-  ( ph  ->  (Vtx `  U )  =  V )
vtxduhgrun.d  |-  ( ph  ->  ( dom  I  i^i 
dom  J )  =  (/) )
vtxduhgrun.g  |-  ( ph  ->  G  e. UHGraph  )
vtxduhgrun.h  |-  ( ph  ->  H  e. UHGraph  )
vtxduhgrun.n  |-  ( ph  ->  N  e.  V )
vtxduhgrun.u  |-  ( ph  ->  (iEdg `  U )  =  ( I  u.  J ) )
Assertion
Ref Expression
vtxduhgrun  |-  ( ph  ->  ( (VtxDeg `  U
) `  N )  =  ( ( (VtxDeg `  G ) `  N
) +e ( (VtxDeg `  H ) `  N ) ) )

Proof of Theorem vtxduhgrun
StepHypRef Expression
1 vtxduhgrun.i . 2  |-  I  =  (iEdg `  G )
2 vtxduhgrun.j . 2  |-  J  =  (iEdg `  H )
3 vtxduhgrun.vg . 2  |-  V  =  (Vtx `  G )
4 vtxduhgrun.vh . 2  |-  ( ph  ->  (Vtx `  H )  =  V )
5 vtxduhgrun.vu . 2  |-  ( ph  ->  (Vtx `  U )  =  V )
6 vtxduhgrun.d . 2  |-  ( ph  ->  ( dom  I  i^i 
dom  J )  =  (/) )
7 vtxduhgrun.g . . 3  |-  ( ph  ->  G  e. UHGraph  )
81uhgrfun 39310 . . 3  |-  ( G  e. UHGraph  ->  Fun  I )
97, 8syl 17 . 2  |-  ( ph  ->  Fun  I )
10 vtxduhgrun.h . . 3  |-  ( ph  ->  H  e. UHGraph  )
112uhgrfun 39310 . . 3  |-  ( H  e. UHGraph  ->  Fun  J )
1210, 11syl 17 . 2  |-  ( ph  ->  Fun  J )
13 vtxduhgrun.n . 2  |-  ( ph  ->  N  e.  V )
14 vtxduhgrun.u . 2  |-  ( ph  ->  (iEdg `  U )  =  ( I  u.  J ) )
151, 2, 3, 4, 5, 6, 9, 12, 13, 14vtxdun 39704 1  |-  ( ph  ->  ( (VtxDeg `  U
) `  N )  =  ( ( (VtxDeg `  G ) `  N
) +e ( (VtxDeg `  H ) `  N ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1452    e. wcel 1904    u. cun 3388    i^i cin 3389   (/)c0 3722   dom cdm 4839   Fun wfun 5583   ` cfv 5589  (class class class)co 6308   +ecxad 11430  Vtxcvtx 39251  iEdgciedg 39252   UHGraph cuhgr 39300  VtxDegcvtxdg 39691
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-rep 4508  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-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-rmo 2764  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-oadd 7204  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-card 8391  df-cda 8616  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-n0 10894  df-z 10962  df-uz 11183  df-xadd 11433  df-hash 12554  df-xnn0 39221  df-uhgr 39302  df-vtxdg 39692
This theorem is referenced by: (None)
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