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Theorem usgedgvadf1ALTlem2 39716
Description: Lemma 2 for usgedgvadf1 39714. (Contributed by Alexander van der Vekens, 4-Jan-2018.) (Revised by AV, 9-Jan-2020.)
Hypotheses
Ref Expression
usgedgvadf1ALT.v  |-  V  =  ( 1st `  G
)
usgedgvadf1ALT.e  |-  E  =  ( Edges  `  G )
usgedgvadf1ALT.a  |-  A  =  { e  e.  E  |  N  e.  e }
Assertion
Ref Expression
usgedgvadf1ALTlem2  |-  ( ( G  e. USGrph  /\  C  e.  A )  ->  ( M  =  ( iota_ m  e.  V  C  =  { m ,  N } )  ->  C  =  { M ,  N } ) )
Distinct variable groups:    C, e    C, m    e, E    m, E    m, G    e, N    m, N    m, V    m, M
Allowed substitution hints:    A( e, m)    G( e)    M( e)    V( e)

Proof of Theorem usgedgvadf1ALTlem2
StepHypRef Expression
1 eleq2 2517 . . . . 5  |-  ( e  =  C  ->  ( N  e.  e  <->  N  e.  C ) )
2 usgedgvadf1ALT.a . . . . 5  |-  A  =  { e  e.  E  |  N  e.  e }
31, 2elrab2 3197 . . . 4  |-  ( C  e.  A  <->  ( C  e.  E  /\  N  e.  C ) )
43biimpi 198 . . 3  |-  ( C  e.  A  ->  ( C  e.  E  /\  N  e.  C )
)
5 simpr 463 . . . . . 6  |-  ( ( ( C  e.  A  /\  ( C  e.  E  /\  N  e.  C
) )  /\  G  e. USGrph  )  ->  G  e. USGrph  )
6 simplrl 769 . . . . . 6  |-  ( ( ( C  e.  A  /\  ( C  e.  E  /\  N  e.  C
) )  /\  G  e. USGrph  )  ->  C  e.  E )
7 simplrr 770 . . . . . 6  |-  ( ( ( C  e.  A  /\  ( C  e.  E  /\  N  e.  C
) )  /\  G  e. USGrph  )  ->  N  e.  C )
8 usgedgvadf1ALT.v . . . . . . 7  |-  V  =  ( 1st `  G
)
9 usgedgvadf1ALT.e . . . . . . 7  |-  E  =  ( Edges  `  G )
108, 9usgvincvadeuALT 39707 . . . . . 6  |-  ( ( G  e. USGrph  /\  C  e.  E  /\  N  e.  C )  ->  E! m  e.  V  C  =  { N ,  m } )
115, 6, 7, 10syl3anc 1267 . . . . 5  |-  ( ( ( C  e.  A  /\  ( C  e.  E  /\  N  e.  C
) )  /\  G  e. USGrph  )  ->  E! m  e.  V  C  =  { N ,  m }
)
12 eqcom 2457 . . . . . . . 8  |-  ( M  =  ( iota_ m  e.  V  C  =  {
m ,  N }
)  <->  ( iota_ m  e.  V  C  =  {
m ,  N }
)  =  M )
138, 9, 2usgedgvadf1ALTlem1 39715 . . . . . . . . . . . . . . . 16  |-  ( ( G  e. USGrph  /\  C  e.  A )  ->  ( iota_ m  e.  V  C  =  { m ,  N } )  e.  V
)
1413expcom 437 . . . . . . . . . . . . . . 15  |-  ( C  e.  A  ->  ( G  e. USGrph  ->  ( iota_ m  e.  V  C  =  { m ,  N } )  e.  V
) )
1514adantr 467 . . . . . . . . . . . . . 14  |-  ( ( C  e.  A  /\  ( C  e.  E  /\  N  e.  C
) )  ->  ( G  e. USGrph  ->  ( iota_ m  e.  V  C  =  { m ,  N } )  e.  V
) )
1615imp 431 . . . . . . . . . . . . 13  |-  ( ( ( C  e.  A  /\  ( C  e.  E  /\  N  e.  C
) )  /\  G  e. USGrph  )  ->  ( iota_ m  e.  V  C  =  { m ,  N } )  e.  V
)
1716adantr 467 . . . . . . . . . . . 12  |-  ( ( ( ( C  e.  A  /\  ( C  e.  E  /\  N  e.  C ) )  /\  G  e. USGrph  )  /\  E! m  e.  V  C  =  { N ,  m } )  ->  ( iota_ m  e.  V  C  =  { m ,  N } )  e.  V
)
1817adantr 467 . . . . . . . . . . 11  |-  ( ( ( ( ( C  e.  A  /\  ( C  e.  E  /\  N  e.  C )
)  /\  G  e. USGrph  )  /\  E! m  e.  V  C  =  { N ,  m }
)  /\  M  =  ( iota_ m  e.  V  C  =  { m ,  N } ) )  ->  ( iota_ m  e.  V  C  =  {
m ,  N }
)  e.  V )
19 eleq1 2516 . . . . . . . . . . . 12  |-  ( M  =  ( iota_ m  e.  V  C  =  {
m ,  N }
)  ->  ( M  e.  V  <->  ( iota_ m  e.  V  C  =  {
m ,  N }
)  e.  V ) )
2019adantl 468 . . . . . . . . . . 11  |-  ( ( ( ( ( C  e.  A  /\  ( C  e.  E  /\  N  e.  C )
)  /\  G  e. USGrph  )  /\  E! m  e.  V  C  =  { N ,  m }
)  /\  M  =  ( iota_ m  e.  V  C  =  { m ,  N } ) )  ->  ( M  e.  V  <->  ( iota_ m  e.  V  C  =  {
m ,  N }
)  e.  V ) )
2118, 20mpbird 236 . . . . . . . . . 10  |-  ( ( ( ( ( C  e.  A  /\  ( C  e.  E  /\  N  e.  C )
)  /\  G  e. USGrph  )  /\  E! m  e.  V  C  =  { N ,  m }
)  /\  M  =  ( iota_ m  e.  V  C  =  { m ,  N } ) )  ->  M  e.  V
)
22 prcom 4049 . . . . . . . . . . . . . . 15  |-  { N ,  m }  =  {
m ,  N }
2322eqeq2i 2462 . . . . . . . . . . . . . 14  |-  ( C  =  { N ,  m }  <->  C  =  {
m ,  N }
)
2423reubii 2976 . . . . . . . . . . . . 13  |-  ( E! m  e.  V  C  =  { N ,  m } 
<->  E! m  e.  V  C  =  { m ,  N } )
2524biimpi 198 . . . . . . . . . . . 12  |-  ( E! m  e.  V  C  =  { N ,  m }  ->  E! m  e.  V  C  =  {
m ,  N }
)
2625adantl 468 . . . . . . . . . . 11  |-  ( ( ( ( C  e.  A  /\  ( C  e.  E  /\  N  e.  C ) )  /\  G  e. USGrph  )  /\  E! m  e.  V  C  =  { N ,  m } )  ->  E! m  e.  V  C  =  { m ,  N } )
2726adantr 467 . . . . . . . . . 10  |-  ( ( ( ( ( C  e.  A  /\  ( C  e.  E  /\  N  e.  C )
)  /\  G  e. USGrph  )  /\  E! m  e.  V  C  =  { N ,  m }
)  /\  M  =  ( iota_ m  e.  V  C  =  { m ,  N } ) )  ->  E! m  e.  V  C  =  {
m ,  N }
)
28 preq1 4050 . . . . . . . . . . . 12  |-  ( m  =  M  ->  { m ,  N }  =  { M ,  N }
)
2928eqeq2d 2460 . . . . . . . . . . 11  |-  ( m  =  M  ->  ( C  =  { m ,  N }  <->  C  =  { M ,  N }
) )
3029riota2 6272 . . . . . . . . . 10  |-  ( ( M  e.  V  /\  E! m  e.  V  C  =  { m ,  N } )  -> 
( C  =  { M ,  N }  <->  (
iota_ m  e.  V  C  =  { m ,  N } )  =  M ) )
3121, 27, 30syl2anc 666 . . . . . . . . 9  |-  ( ( ( ( ( C  e.  A  /\  ( C  e.  E  /\  N  e.  C )
)  /\  G  e. USGrph  )  /\  E! m  e.  V  C  =  { N ,  m }
)  /\  M  =  ( iota_ m  e.  V  C  =  { m ,  N } ) )  ->  ( C  =  { M ,  N } 
<->  ( iota_ m  e.  V  C  =  { m ,  N } )  =  M ) )
3231biimprd 227 . . . . . . . 8  |-  ( ( ( ( ( C  e.  A  /\  ( C  e.  E  /\  N  e.  C )
)  /\  G  e. USGrph  )  /\  E! m  e.  V  C  =  { N ,  m }
)  /\  M  =  ( iota_ m  e.  V  C  =  { m ,  N } ) )  ->  ( ( iota_ m  e.  V  C  =  { m ,  N } )  =  M  ->  C  =  { M ,  N }
) )
3312, 32syl5bi 221 . . . . . . 7  |-  ( ( ( ( ( C  e.  A  /\  ( C  e.  E  /\  N  e.  C )
)  /\  G  e. USGrph  )  /\  E! m  e.  V  C  =  { N ,  m }
)  /\  M  =  ( iota_ m  e.  V  C  =  { m ,  N } ) )  ->  ( M  =  ( iota_ m  e.  V  C  =  { m ,  N } )  ->  C  =  { M ,  N } ) )
3433ex 436 . . . . . 6  |-  ( ( ( ( C  e.  A  /\  ( C  e.  E  /\  N  e.  C ) )  /\  G  e. USGrph  )  /\  E! m  e.  V  C  =  { N ,  m } )  ->  ( M  =  ( iota_ m  e.  V  C  =  { m ,  N } )  ->  ( M  =  ( iota_ m  e.  V  C  =  { m ,  N } )  ->  C  =  { M ,  N } ) ) )
3534pm2.43d 50 . . . . 5  |-  ( ( ( ( C  e.  A  /\  ( C  e.  E  /\  N  e.  C ) )  /\  G  e. USGrph  )  /\  E! m  e.  V  C  =  { N ,  m } )  ->  ( M  =  ( iota_ m  e.  V  C  =  { m ,  N } )  ->  C  =  { M ,  N } ) )
3611, 35mpdan 673 . . . 4  |-  ( ( ( C  e.  A  /\  ( C  e.  E  /\  N  e.  C
) )  /\  G  e. USGrph  )  ->  ( M  =  ( iota_ m  e.  V  C  =  {
m ,  N }
)  ->  C  =  { M ,  N }
) )
3736ex 436 . . 3  |-  ( ( C  e.  A  /\  ( C  e.  E  /\  N  e.  C
) )  ->  ( G  e. USGrph  ->  ( M  =  ( iota_ m  e.  V  C  =  {
m ,  N }
)  ->  C  =  { M ,  N }
) ) )
384, 37mpdan 673 . 2  |-  ( C  e.  A  ->  ( G  e. USGrph  ->  ( M  =  ( iota_ m  e.  V  C  =  {
m ,  N }
)  ->  C  =  { M ,  N }
) ) )
3938impcom 432 1  |-  ( ( G  e. USGrph  /\  C  e.  A )  ->  ( M  =  ( iota_ m  e.  V  C  =  { m ,  N } )  ->  C  =  { M ,  N } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    = wceq 1443    e. wcel 1886   E!wreu 2738   {crab 2740   {cpr 3969   ` cfv 5581   iota_crio 6249   1stc1st 6788   USGrph cusg 25050   Edges cedg 25051
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580  ax-cnex 9592  ax-resscn 9593  ax-1cn 9594  ax-icn 9595  ax-addcl 9596  ax-addrcl 9597  ax-mulcl 9598  ax-mulrcl 9599  ax-mulcom 9600  ax-addass 9601  ax-mulass 9602  ax-distr 9603  ax-i2m1 9604  ax-1ne0 9605  ax-1rid 9606  ax-rnegex 9607  ax-rrecex 9608  ax-cnre 9609  ax-pre-lttri 9610  ax-pre-lttrn 9611  ax-pre-ltadd 9612  ax-pre-mulgt0 9613
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rmo 2744  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-riota 6250  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-1st 6790  df-2nd 6791  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-2o 7180  df-oadd 7183  df-er 7360  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-card 8370  df-cda 8595  df-pnf 9674  df-mnf 9675  df-xr 9676  df-ltxr 9677  df-le 9678  df-sub 9859  df-neg 9860  df-nn 10607  df-2 10665  df-n0 10867  df-z 10935  df-uz 11157  df-fz 11782  df-hash 12513  df-usgra 25053  df-edg 25056
This theorem is referenced by:  usgedgvadf1ALT  39717
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