Theorem upgrm 15746

Description: An edge is an inhabited subset of vertices. (Contributed by Mario Carneiro, 11-Mar-2015.) (Revised by AV, 10-Oct-2020.)

Hypotheses

Ref	Expression
isupgr.v	|- V = (Vtx ` G )
isupgr.e	|- E = (iEdg ` G )

Assertion

Ref	Expression
upgrm	|- ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> E. j j e. ( E ` F ) )

Distinct variable groups:   j, E   j, F

Allowed substitution hints:   A( j)   G( j)   V( j)

Proof of Theorem upgrm

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		isupgr.v	. . . . . 6 |- V = (Vtx ` G )
2		isupgr.e	. . . . . 6 |- E = (iEdg ` G )
3	1, 2	upgrfnen 15744	. . . . 5 |- ( ( G e. UPGraph /\ E Fn A ) -> E : A --> { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } )
4	3	ffvelcdmda 5725	. . . 4 |- ( ( ( G e. UPGraph /\ E Fn A ) /\ F e. A ) -> ( E ` F ) e. { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } )
5	4	3impa 1197	. . 3 |- ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> ( E ` F ) e. { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } )
6		breq1 4051	. . . . 5 |- ( x = ( E ` F ) -> ( x ~~ 1o <-> ( E ` F ) ~~ 1o ) )
7		breq1 4051	. . . . 5 |- ( x = ( E ` F ) -> ( x ~~ 2o <-> ( E ` F ) ~~ 2o ) )
8	6, 7	orbi12d 795	. . . 4 |- ( x = ( E ` F ) -> ( ( x ~~ 1o \/ x ~~ 2o ) <-> ( ( E ` F ) ~~ 1o \/ ( E ` F ) ~~ 2o ) ) )
9	8	elrab 2931	. . 3 |- ( ( E ` F ) e. { x e. ~P V | ( x ~~ 1o \/ x ~~ 2o ) } <-> ( ( E ` F ) e. ~P V /\ ( ( E ` F ) ~~ 1o \/ ( E ` F ) ~~ 2o ) ) )
10	5, 9	sylib 122	. 2 |- ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> ( ( E ` F ) e. ~P V /\ ( ( E ` F ) ~~ 1o \/ ( E ` F ) ~~ 2o ) ) )
11		en1m 6907	. . 3 |- ( ( E ` F ) ~~ 1o -> E. j j e. ( E ` F ) )
12		en2m 6924	. . 3 |- ( ( E ` F ) ~~ 2o -> E. j j e. ( E ` F ) )
13	11, 12	jaoi 718	. 2 |- ( ( ( E ` F ) ~~ 1o \/ ( E ` F ) ~~ 2o ) -> E. j j e. ( E ` F ) )
14	10, 13	simpl2im 386	1 |- ( ( G e. UPGraph /\ E Fn A /\ F e. A ) -> E. j j e. ( E ` F ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 710   /\ w3a 981   = wceq 1373   E.wex 1516   e. wcel 2177   {crab 2489   ~Pcpw 3618   class class class wbr 4048   Fn wfn 5272   ` cfv 5277   1oc1o 6505   2oc2o 6506   ~~ cen 6835  Vtxcvtx 15661  iEdgciedg 15662  UPGraphcupgr 15737

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4167  ax-nul 4175  ax-pow 4223  ax-pr 4258  ax-un 4485  ax-setind 4590  ax-cnex 8029  ax-resscn 8030  ax-1cn 8031  ax-1re 8032  ax-icn 8033  ax-addcl 8034  ax-addrcl 8035  ax-mulcl 8036  ax-addcom 8038  ax-mulcom 8039  ax-addass 8040  ax-mulass 8041  ax-distr 8042  ax-i2m1 8043  ax-1rid 8045  ax-0id 8046  ax-rnegex 8047  ax-cnre 8049

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3001  df-csb 3096  df-dif 3170  df-un 3172  df-in 3174  df-ss 3181  df-nul 3463  df-if 3574  df-pw 3620  df-sn 3641  df-pr 3642  df-op 3644  df-uni 3854  df-int 3889  df-br 4049  df-opab 4111  df-mpt 4112  df-tr 4148  df-id 4345  df-iord 4418  df-on 4420  df-suc 4423  df-xp 4686  df-rel 4687  df-cnv 4688  df-co 4689  df-dm 4690  df-rn 4691  df-res 4692  df-ima 4693  df-iota 5238  df-fun 5279  df-fn 5280  df-f 5281  df-f1 5282  df-fo 5283  df-f1o 5284  df-fv 5285  df-riota 5909  df-ov 5957  df-oprab 5958  df-mpo 5959  df-1st 6236  df-2nd 6237  df-1o 6512  df-2o 6513  df-en 6838  df-sub 8258  df-inn 9050  df-2 9108  df-3 9109  df-4 9110  df-5 9111  df-6 9112  df-7 9113  df-8 9114  df-9 9115  df-n0 9309  df-dec 9518  df-ndx 12885  df-slot 12886  df-base 12888  df-edgf 15654  df-vtx 15663  df-iedg 15664  df-upgren 15739

This theorem is referenced by: (None)