Theorem usgrsizedgen 16337

Description: In a simple graph, the size of the edge function is the number of the edges of the graph. (Contributed by AV, 4-Jan-2020.) (Revised by AV, 7-Jun-2021.)

Assertion

Ref	Expression
usgrsizedgen	|- ( G e. USGraph -> (iEdg ` G ) ~~ (Edg ` G ) )

Proof of Theorem usgrsizedgen

Step	Hyp	Ref	Expression
1		iedgex 16143	. . . . 5 |- ( G e. USGraph -> (iEdg ` G ) e. _V )
2		usgrfun 16285	. . . . 5 |- ( G e. USGraph -> Fun (iEdg ` G ) )
3		fundmeng 7061	. . . . 5 |- ( ( (iEdg ` G ) e. _V /\ Fun (iEdg ` G ) ) -> dom (iEdg ` G ) ~~ (iEdg ` G ) )
4	1, 2, 3	syl2anc 411	. . . 4 |- ( G e. USGraph -> dom (iEdg ` G ) ~~ (iEdg ` G ) )
5	4	ensymd 7036	. . 3 |- ( G e. USGraph -> (iEdg ` G ) ~~ dom (iEdg ` G ) )
6	1	dmexd 5028	. . . 4 |- ( G e. USGraph -> dom (iEdg ` G ) e. _V )
7		eqid 2234	. . . . 5 |- (iEdg ` G ) = (iEdg ` G )
8	7	usgrf1o 16298	. . . 4 |- ( G e. USGraph -> (iEdg ` G ) : dom (iEdg ` G ) -1-1-onto-> ran (iEdg ` G ) )
9		f1oeng 7009	. . . 4 |- ( ( dom (iEdg ` G ) e. _V /\ (iEdg ` G ) : dom (iEdg ` G ) -1-1-onto-> ran (iEdg ` G ) ) -> dom (iEdg ` G ) ~~ ran (iEdg ` G ) )
10	6, 8, 9	syl2anc 411	. . 3 |- ( G e. USGraph -> dom (iEdg ` G ) ~~ ran (iEdg ` G ) )
11		entr 7037	. . 3 |- ( ( (iEdg ` G ) ~~ dom (iEdg ` G ) /\ dom (iEdg ` G ) ~~ ran (iEdg ` G ) ) -> (iEdg ` G ) ~~ ran (iEdg ` G ) )
12	5, 10, 11	syl2anc 411	. 2 |- ( G e. USGraph -> (iEdg ` G ) ~~ ran (iEdg ` G ) )
13		edgvalg 16183	. 2 |- ( G e. USGraph -> (Edg ` G ) = ran (iEdg ` G ) )
14	12, 13	breqtrrd 4142	1 |- ( G e. USGraph -> (iEdg ` G ) ~~ (Edg ` G ) )

Syntax hints:   -> wi 4   e. wcel 2205   _Vcvv 2815   class class class wbr 4114   dom cdm 4754   ran crn 4755   Fun wfun 5351   -1-1-onto->wf1o 5356   ` cfv 5357   ~~ cen 6986  iEdgciedg 16137  Edgcedg 16181  USGraphcusgr 16278

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-1cn 8236  ax-1re 8237  ax-icn 8238  ax-addcl 8239  ax-addrcl 8240  ax-mulcl 8241  ax-addcom 8243  ax-mulcom 8244  ax-addass 8245  ax-mulass 8246  ax-distr 8247  ax-i2m1 8248  ax-1rid 8250  ax-0id 8251  ax-rnegex 8252  ax-cnre 8254

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-riota 6011  df-ov 6061  df-oprab 6062  df-mpo 6063  df-1st 6347  df-2nd 6348  df-er 6780  df-en 6989  df-sub 8463  df-inn 9258  df-2 9316  df-3 9317  df-4 9318  df-5 9319  df-6 9320  df-7 9321  df-8 9322  df-9 9323  df-n0 9517  df-dec 9731  df-ndx 13302  df-slot 13303  df-base 13305  df-edgf 16129  df-vtx 16138  df-iedg 16139  df-edg 16182  df-usgren 16280

This theorem is referenced by: (None)