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Theorem umgraun 21355
 Description: If and are graphs, then is a graph (the vertex set stays the same, but the edges from both graphs are kept). (Contributed by Mario Carneiro, 12-Mar-2015.)
Hypotheses
Ref Expression
umgraun.e
umgraun.f
umgraun.i
umgraun.ge UMGrph
umgraun.gf UMGrph
Assertion
Ref Expression
umgraun UMGrph

Proof of Theorem umgraun
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 umgraun.ge . . . . 5 UMGrph
2 umgraun.e . . . . 5
3 umgraf 21345 . . . . 5 UMGrph
41, 2, 3syl2anc 643 . . . 4
5 umgraun.gf . . . . 5 UMGrph
6 umgraun.f . . . . 5
7 umgraf 21345 . . . . 5 UMGrph
85, 6, 7syl2anc 643 . . . 4
9 umgraun.i . . . 4
10 fun2 5600 . . . 4
114, 8, 9, 10syl21anc 1183 . . 3
12 fdm 5587 . . . . 5
1311, 12syl 16 . . . 4
1413feq2d 5573 . . 3
1511, 14mpbird 224 . 2
16 relumgra 21341 . . . 4 UMGrph
17 releldm 5094 . . . 4 UMGrph UMGrph UMGrph
1816, 1, 17sylancr 645 . . 3 UMGrph
1916brrelex2i 4911 . . . . 5 UMGrph
201, 19syl 16 . . . 4
2116brrelex2i 4911 . . . . 5 UMGrph
225, 21syl 16 . . . 4
23 unexg 4702 . . . 4
2420, 22, 23syl2anc 643 . . 3
25 isumgra 21342 . . 3 UMGrph UMGrph
2618, 24, 25syl2anc 643 . 2 UMGrph
2715, 26mpbird 224 1 UMGrph
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wceq 1652   wcel 1725  crab 2701  cvv 2948   cdif 3309   cun 3310   cin 3311  c0 3620  cpw 3791  csn 3806   class class class wbr 4204   cdm 4870   wrel 4875   wfn 5441  wf 5442  cfv 5446   cle 9113  c2 10041  chash 11610   UMGrph cumg 21339 This theorem is referenced by:  uslgraun  21386  eupap1  21690 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-un 4693 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-fun 5448  df-fn 5449  df-f 5450  df-umgra 21340
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