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Theorem usgra0v 24144
 Description: The empty graph with no vertices is a graph if and only if the edge function is empty. (Contributed by Alexander van der Vekens, 30-Sep-2017.)
Assertion
Ref Expression
usgra0v USGrph

Proof of Theorem usgra0v
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 usgrav 24111 . . 3 USGrph
2 isusgra 24117 . . . 4 USGrph
3 eqidd 2468 . . . . . . 7
4 eqidd 2468 . . . . . . 7
5 pw0 4174 . . . . . . . . . . . 12
65difeq1i 3618 . . . . . . . . . . 11
7 difid 3895 . . . . . . . . . . 11
86, 7eqtri 2496 . . . . . . . . . 10
98a1i 11 . . . . . . . . 9
10 biidd 237 . . . . . . . . 9
119, 10rabeqbidv 3108 . . . . . . . 8
12 rab0 3806 . . . . . . . 8
1311, 12syl6eq 2524 . . . . . . 7
143, 4, 13f1eq123d 5811 . . . . . 6
15 f1f 5781 . . . . . . 7
16 f00 5767 . . . . . . . 8
1716simplbi 460 . . . . . . 7
1815, 17syl 16 . . . . . 6
1914, 18syl6bi 228 . . . . 5
2019adantl 466 . . . 4
212, 20sylbid 215 . . 3 USGrph
221, 21mpcom 36 . 2 USGrph
23 0ex 4577 . . . 4
24 usgra0 24143 . . . 4 USGrph
2523, 24ax-mp 5 . . 3 USGrph
26 breq2 4451 . . 3 USGrph USGrph
2725, 26mpbiri 233 . 2 USGrph
2822, 27impbii 188 1 USGrph
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 184   wa 369   wceq 1379   wcel 1767  crab 2818  cvv 3113   cdif 3473  c0 3785  cpw 4010  csn 4027   class class class wbr 4447   cdm 4999  wf 5584  wf1 5585  cfv 5588  c2 10586  chash 12374   USGrph cusg 24103 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-usgra 24106 This theorem is referenced by:  usgra1v  24163  usgrafisindb0  24181  frgra0v  24766  usgo0s0  32129  usgo0s0ALT  32130
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