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Theorem usgra0v 21396
 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 21376 . . 3 USGrph
2 isusgra 21378 . . . 4 USGrph
3 eqidd 2439 . . . . . . 7
4 eqidd 2439 . . . . . . 7
5 pw0 3947 . . . . . . . . . . . 12
65difeq1i 3463 . . . . . . . . . . 11
7 difid 3698 . . . . . . . . . . 11
86, 7eqtri 2458 . . . . . . . . . 10
98a1i 11 . . . . . . . . 9
10 biidd 230 . . . . . . . . 9
119, 10rabeqbidv 2953 . . . . . . . 8
12 rab0 3650 . . . . . . . 8
1311, 12syl6eq 2486 . . . . . . 7
143, 4, 13f1eq123d 5672 . . . . . 6
15 f1f 5642 . . . . . . 7
16 f00 5631 . . . . . . . 8
1716simplbi 448 . . . . . . 7
1815, 17syl 16 . . . . . 6
1914, 18syl6bi 221 . . . . 5
2019adantl 454 . . . 4
212, 20sylbid 208 . . 3 USGrph
221, 21mpcom 35 . 2 USGrph
23 0ex 4342 . . . 4
24 usgra0 21395 . . . 4 USGrph
2523, 24ax-mp 5 . . 3 USGrph
26 breq2 4219 . . 3 USGrph USGrph
2725, 26mpbiri 226 . 2 USGrph
2822, 27impbii 182 1 USGrph
 Colors of variables: wff set class Syntax hints:   wi 4   wb 178   wa 360   wceq 1653   wcel 1726  crab 2711  cvv 2958   cdif 3319  c0 3630  cpw 3801  csn 3816   class class class wbr 4215   cdm 4881  wf 5453  wf1 5454  cfv 5457  c2 10054  chash 11623   USGrph cusg 21370 This theorem is referenced by:  usgra1v  21414  usgrafisindb0  21427  frgra0v  28457 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-opab 4270  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-usgra 21372
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