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Theorem 3vfriswmgralem 28428
Description: Lemma for 3vfriswmgra 28429. (Contributed by Alexander van der Vekens, 6-Oct-2017.)
Assertion
Ref Expression
3vfriswmgralem  |-  ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  ->  ( { A ,  B }  e.  ran  E  ->  E! w  e.  { A ,  B }  { A ,  w }  e.  ran  E ) )
Distinct variable groups:    w, A    w, B    w, C    w, E    w, X    w, Y

Proof of Theorem 3vfriswmgralem
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 simpr 447 . . . . . . 7  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  { A ,  B }  e.  ran  E )
21olcd 382 . . . . . 6  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  ( { A ,  A }  e.  ran  E  \/  { A ,  B }  e.  ran  E ) )
3 preq2 3720 . . . . . . . . . 10  |-  ( w  =  A  ->  { A ,  w }  =  { A ,  A }
)
43eleq1d 2362 . . . . . . . . 9  |-  ( w  =  A  ->  ( { A ,  w }  e.  ran  E  <->  { A ,  A }  e.  ran  E ) )
5 preq2 3720 . . . . . . . . . 10  |-  ( w  =  B  ->  { A ,  w }  =  { A ,  B }
)
65eleq1d 2362 . . . . . . . . 9  |-  ( w  =  B  ->  ( { A ,  w }  e.  ran  E  <->  { A ,  B }  e.  ran  E ) )
74, 6rexprg 3696 . . . . . . . 8  |-  ( ( A  e.  X  /\  B  e.  Y )  ->  ( E. w  e. 
{ A ,  B }  { A ,  w }  e.  ran  E  <->  ( { A ,  A }  e.  ran  E  \/  { A ,  B }  e.  ran  E ) ) )
873ad2ant1 976 . . . . . . 7  |-  ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  ->  ( E. w  e.  { A ,  B }  { A ,  w }  e.  ran  E  <-> 
( { A ,  A }  e.  ran  E  \/  { A ,  B }  e.  ran  E ) ) )
98adantr 451 . . . . . 6  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  ( E. w  e.  { A ,  B }  { A ,  w }  e.  ran  E  <-> 
( { A ,  A }  e.  ran  E  \/  { A ,  B }  e.  ran  E ) ) )
102, 9mpbird 223 . . . . 5  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  E. w  e.  { A ,  B }  { A ,  w }  e.  ran  E )
11 df-rex 2562 . . . . 5  |-  ( E. w  e.  { A ,  B }  { A ,  w }  e.  ran  E  <->  E. w ( w  e. 
{ A ,  B }  /\  { A ,  w }  e.  ran  E ) )
1210, 11sylib 188 . . . 4  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  E. w
( w  e.  { A ,  B }  /\  { A ,  w }  e.  ran  E ) )
13 vex 2804 . . . . . . . . 9  |-  w  e. 
_V
1413elpr 3671 . . . . . . . 8  |-  ( w  e.  { A ,  B }  <->  ( w  =  A  \/  w  =  B ) )
15 vex 2804 . . . . . . . . . . . 12  |-  y  e. 
_V
1615elpr 3671 . . . . . . . . . . 11  |-  ( y  e.  { A ,  B }  <->  ( y  =  A  \/  y  =  B ) )
17 eqidd 2297 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  A  =  A )
1817a1i 10 . . . . . . . . . . . . . . . . 17  |-  ( { A ,  A }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  A  =  A ) )
1918a1ii 24 . . . . . . . . . . . . . . . 16  |-  ( y  =  A  ->  ( { A ,  A }  e.  ran  E  ->  ( { A ,  A }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  A  =  A ) ) ) )
20 preq2 3720 . . . . . . . . . . . . . . . . 17  |-  ( y  =  A  ->  { A ,  y }  =  { A ,  A }
)
2120eleq1d 2362 . . . . . . . . . . . . . . . 16  |-  ( y  =  A  ->  ( { A ,  y }  e.  ran  E  <->  { A ,  A }  e.  ran  E ) )
22 eqeq2 2305 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  A  ->  ( A  =  y  <->  A  =  A ) )
2322imbi2d 307 . . . . . . . . . . . . . . . . 17  |-  ( y  =  A  ->  (
( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  A  =  y )  <->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  A  =  A ) ) )
2423imbi2d 307 . . . . . . . . . . . . . . . 16  |-  ( y  =  A  ->  (
( { A ,  A }  e.  ran  E  ->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  A  =  y ) )  <->  ( { A ,  A }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  A  =  A ) ) ) )
2519, 21, 243imtr4d 259 . . . . . . . . . . . . . . 15  |-  ( y  =  A  ->  ( { A ,  y }  e.  ran  E  -> 
( { A ,  A }  e.  ran  E  ->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  A  =  y ) ) ) )
26 usgraedgrn 28259 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( { A ,  B ,  C } USGrph  E  /\  { A ,  A }  e.  ran  E )  ->  A  =/=  A )
27 df-ne 2461 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( A  =/=  A  <->  -.  A  =  A )
28 eqid 2296 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  A  =  A
2928pm2.24i 136 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( -.  A  =  A  ->  A  =  B )
3027, 29sylbi 187 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( A  =/=  A  ->  A  =  B )
3126, 30syl 15 . . . . . . . . . . . . . . . . . . . . 21  |-  ( ( { A ,  B ,  C } USGrph  E  /\  { A ,  A }  e.  ran  E )  ->  A  =  B )
3231ex 423 . . . . . . . . . . . . . . . . . . . 20  |-  ( { A ,  B ,  C } USGrph  E  ->  ( { A ,  A }  e.  ran  E  ->  A  =  B ) )
33323ad2ant3 978 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  ->  ( { A ,  A }  e.  ran  E  ->  A  =  B ) )
3433adantr 451 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  ( { A ,  A }  e.  ran  E  ->  A  =  B ) )
3534com12 27 . . . . . . . . . . . . . . . . 17  |-  ( { A ,  A }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  A  =  B ) )
3635a1ii 24 . . . . . . . . . . . . . . . 16  |-  ( y  =  B  ->  ( { A ,  B }  e.  ran  E  ->  ( { A ,  A }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  A  =  B ) ) ) )
37 preq2 3720 . . . . . . . . . . . . . . . . 17  |-  ( y  =  B  ->  { A ,  y }  =  { A ,  B }
)
3837eleq1d 2362 . . . . . . . . . . . . . . . 16  |-  ( y  =  B  ->  ( { A ,  y }  e.  ran  E  <->  { A ,  B }  e.  ran  E ) )
39 eqeq2 2305 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  B  ->  ( A  =  y  <->  A  =  B ) )
4039imbi2d 307 . . . . . . . . . . . . . . . . 17  |-  ( y  =  B  ->  (
( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  A  =  y )  <->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  A  =  B ) ) )
4140imbi2d 307 . . . . . . . . . . . . . . . 16  |-  ( y  =  B  ->  (
( { A ,  A }  e.  ran  E  ->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  A  =  y ) )  <->  ( { A ,  A }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  A  =  B ) ) ) )
4236, 38, 413imtr4d 259 . . . . . . . . . . . . . . 15  |-  ( y  =  B  ->  ( { A ,  y }  e.  ran  E  -> 
( { A ,  A }  e.  ran  E  ->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  A  =  y ) ) ) )
4325, 42jaoi 368 . . . . . . . . . . . . . 14  |-  ( ( y  =  A  \/  y  =  B )  ->  ( { A , 
y }  e.  ran  E  ->  ( { A ,  A }  e.  ran  E  ->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  A  =  y ) ) ) )
44 eqeq1 2302 . . . . . . . . . . . . . . . . 17  |-  ( w  =  A  ->  (
w  =  y  <->  A  =  y ) )
4544imbi2d 307 . . . . . . . . . . . . . . . 16  |-  ( w  =  A  ->  (
( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  w  =  y )  <->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  A  =  y ) ) )
464, 45imbi12d 311 . . . . . . . . . . . . . . 15  |-  ( w  =  A  ->  (
( { A ,  w }  e.  ran  E  ->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  w  =  y ) )  <->  ( { A ,  A }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  A  =  y ) ) ) )
4746imbi2d 307 . . . . . . . . . . . . . 14  |-  ( w  =  A  ->  (
( { A , 
y }  e.  ran  E  ->  ( { A ,  w }  e.  ran  E  ->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  w  =  y ) ) )  <-> 
( { A , 
y }  e.  ran  E  ->  ( { A ,  A }  e.  ran  E  ->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  A  =  y ) ) ) ) )
4843, 47syl5ibr 212 . . . . . . . . . . . . 13  |-  ( w  =  A  ->  (
( y  =  A  \/  y  =  B )  ->  ( { A ,  y }  e.  ran  E  ->  ( { A ,  w }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  w  =  y ) ) ) ) )
4928pm2.24i 136 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( -.  A  =  A  ->  B  =  A )
5027, 49sylbi 187 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( A  =/=  A  ->  B  =  A )
5126, 50syl 15 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( ( { A ,  B ,  C } USGrph  E  /\  { A ,  A }  e.  ran  E )  ->  B  =  A )
5251ex 423 . . . . . . . . . . . . . . . . . . . . 21  |-  ( { A ,  B ,  C } USGrph  E  ->  ( { A ,  A }  e.  ran  E  ->  B  =  A ) )
53523ad2ant3 978 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  ->  ( { A ,  A }  e.  ran  E  ->  B  =  A ) )
5453adantr 451 . . . . . . . . . . . . . . . . . . 19  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  ( { A ,  A }  e.  ran  E  ->  B  =  A ) )
5554com12 27 . . . . . . . . . . . . . . . . . 18  |-  ( { A ,  A }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  B  =  A ) )
5655a1d 22 . . . . . . . . . . . . . . . . 17  |-  ( { A ,  A }  e.  ran  E  ->  ( { A ,  B }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  B  =  A ) ) )
5756a1i 10 . . . . . . . . . . . . . . . 16  |-  ( y  =  A  ->  ( { A ,  A }  e.  ran  E  ->  ( { A ,  B }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  B  =  A ) ) ) )
58 eqeq2 2305 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  A  ->  ( B  =  y  <->  B  =  A ) )
5958imbi2d 307 . . . . . . . . . . . . . . . . 17  |-  ( y  =  A  ->  (
( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  B  =  y )  <->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  B  =  A ) ) )
6059imbi2d 307 . . . . . . . . . . . . . . . 16  |-  ( y  =  A  ->  (
( { A ,  B }  e.  ran  E  ->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  B  =  y ) )  <->  ( { A ,  B }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  B  =  A ) ) ) )
6157, 21, 603imtr4d 259 . . . . . . . . . . . . . . 15  |-  ( y  =  A  ->  ( { A ,  y }  e.  ran  E  -> 
( { A ,  B }  e.  ran  E  ->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  B  =  y ) ) ) )
62 eqidd 2297 . . . . . . . . . . . . . . . . . 18  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  B  =  B )
6362a1i 10 . . . . . . . . . . . . . . . . 17  |-  ( { A ,  B }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  B  =  B ) )
6463a1ii 24 . . . . . . . . . . . . . . . 16  |-  ( y  =  B  ->  ( { A ,  B }  e.  ran  E  ->  ( { A ,  B }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  B  =  B ) ) ) )
65 eqeq2 2305 . . . . . . . . . . . . . . . . . 18  |-  ( y  =  B  ->  ( B  =  y  <->  B  =  B ) )
6665imbi2d 307 . . . . . . . . . . . . . . . . 17  |-  ( y  =  B  ->  (
( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  B  =  y )  <->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  B  =  B ) ) )
6766imbi2d 307 . . . . . . . . . . . . . . . 16  |-  ( y  =  B  ->  (
( { A ,  B }  e.  ran  E  ->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  B  =  y ) )  <->  ( { A ,  B }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  B  =  B ) ) ) )
6864, 38, 673imtr4d 259 . . . . . . . . . . . . . . 15  |-  ( y  =  B  ->  ( { A ,  y }  e.  ran  E  -> 
( { A ,  B }  e.  ran  E  ->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  B  =  y ) ) ) )
6961, 68jaoi 368 . . . . . . . . . . . . . 14  |-  ( ( y  =  A  \/  y  =  B )  ->  ( { A , 
y }  e.  ran  E  ->  ( { A ,  B }  e.  ran  E  ->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  B  =  y ) ) ) )
70 eqeq1 2302 . . . . . . . . . . . . . . . . 17  |-  ( w  =  B  ->  (
w  =  y  <->  B  =  y ) )
7170imbi2d 307 . . . . . . . . . . . . . . . 16  |-  ( w  =  B  ->  (
( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  w  =  y )  <->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  B  =  y ) ) )
726, 71imbi12d 311 . . . . . . . . . . . . . . 15  |-  ( w  =  B  ->  (
( { A ,  w }  e.  ran  E  ->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  w  =  y ) )  <->  ( { A ,  B }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  B  =  y ) ) ) )
7372imbi2d 307 . . . . . . . . . . . . . 14  |-  ( w  =  B  ->  (
( { A , 
y }  e.  ran  E  ->  ( { A ,  w }  e.  ran  E  ->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  w  =  y ) ) )  <-> 
( { A , 
y }  e.  ran  E  ->  ( { A ,  B }  e.  ran  E  ->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  B  =  y ) ) ) ) )
7469, 73syl5ibr 212 . . . . . . . . . . . . 13  |-  ( w  =  B  ->  (
( y  =  A  \/  y  =  B )  ->  ( { A ,  y }  e.  ran  E  ->  ( { A ,  w }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  w  =  y ) ) ) ) )
7548, 74jaoi 368 . . . . . . . . . . . 12  |-  ( ( w  =  A  \/  w  =  B )  ->  ( ( y  =  A  \/  y  =  B )  ->  ( { A ,  y }  e.  ran  E  -> 
( { A ,  w }  e.  ran  E  ->  ( ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  w  =  y ) ) ) ) )
7675com3l 75 . . . . . . . . . . 11  |-  ( ( y  =  A  \/  y  =  B )  ->  ( { A , 
y }  e.  ran  E  ->  ( ( w  =  A  \/  w  =  B )  ->  ( { A ,  w }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  w  =  y ) ) ) ) )
7716, 76sylbi 187 . . . . . . . . . 10  |-  ( y  e.  { A ,  B }  ->  ( { A ,  y }  e.  ran  E  -> 
( ( w  =  A  \/  w  =  B )  ->  ( { A ,  w }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  w  =  y ) ) ) ) )
7877imp 418 . . . . . . . . 9  |-  ( ( y  e.  { A ,  B }  /\  { A ,  y }  e.  ran  E )  -> 
( ( w  =  A  \/  w  =  B )  ->  ( { A ,  w }  e.  ran  E  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  w  =  y ) ) ) )
7978com3l 75 . . . . . . . 8  |-  ( ( w  =  A  \/  w  =  B )  ->  ( { A ,  w }  e.  ran  E  ->  ( ( y  e.  { A ,  B }  /\  { A ,  y }  e.  ran  E )  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  w  =  y ) ) ) )
8014, 79sylbi 187 . . . . . . 7  |-  ( w  e.  { A ,  B }  ->  ( { A ,  w }  e.  ran  E  ->  (
( y  e.  { A ,  B }  /\  { A ,  y }  e.  ran  E
)  ->  ( (
( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  w  =  y ) ) ) )
8180imp31 421 . . . . . 6  |-  ( ( ( w  e.  { A ,  B }  /\  { A ,  w }  e.  ran  E )  /\  ( y  e. 
{ A ,  B }  /\  { A , 
y }  e.  ran  E ) )  ->  (
( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  w  =  y ) )
8281com12 27 . . . . 5  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  ( (
( w  e.  { A ,  B }  /\  { A ,  w }  e.  ran  E )  /\  ( y  e. 
{ A ,  B }  /\  { A , 
y }  e.  ran  E ) )  ->  w  =  y ) )
8382alrimivv 1622 . . . 4  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  A. w A. y ( ( ( w  e.  { A ,  B }  /\  { A ,  w }  e.  ran  E )  /\  ( y  e.  { A ,  B }  /\  { A ,  y }  e.  ran  E
) )  ->  w  =  y ) )
84 eleq1 2356 . . . . . 6  |-  ( w  =  y  ->  (
w  e.  { A ,  B }  <->  y  e.  { A ,  B }
) )
85 preq2 3720 . . . . . . 7  |-  ( w  =  y  ->  { A ,  w }  =  { A ,  y }
)
8685eleq1d 2362 . . . . . 6  |-  ( w  =  y  ->  ( { A ,  w }  e.  ran  E  <->  { A ,  y }  e.  ran  E ) )
8784, 86anbi12d 691 . . . . 5  |-  ( w  =  y  ->  (
( w  e.  { A ,  B }  /\  { A ,  w }  e.  ran  E )  <-> 
( y  e.  { A ,  B }  /\  { A ,  y }  e.  ran  E
) ) )
8887eu4 2195 . . . 4  |-  ( E! w ( w  e. 
{ A ,  B }  /\  { A ,  w }  e.  ran  E )  <->  ( E. w
( w  e.  { A ,  B }  /\  { A ,  w }  e.  ran  E )  /\  A. w A. y ( ( ( w  e.  { A ,  B }  /\  { A ,  w }  e.  ran  E )  /\  ( y  e.  { A ,  B }  /\  { A ,  y }  e.  ran  E
) )  ->  w  =  y ) ) )
8912, 83, 88sylanbrc 645 . . 3  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  E! w
( w  e.  { A ,  B }  /\  { A ,  w }  e.  ran  E ) )
90 df-reu 2563 . . 3  |-  ( E! w  e.  { A ,  B }  { A ,  w }  e.  ran  E  <-> 
E! w ( w  e.  { A ,  B }  /\  { A ,  w }  e.  ran  E ) )
9189, 90sylibr 203 . 2  |-  ( ( ( ( A  e.  X  /\  B  e.  Y )  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  /\  { A ,  B }  e.  ran  E )  ->  E! w  e.  { A ,  B }  { A ,  w }  e.  ran  E )
9291ex 423 1  |-  ( ( ( A  e.  X  /\  B  e.  Y
)  /\  A  =/=  B  /\  { A ,  B ,  C } USGrph  E )  ->  ( { A ,  B }  e.  ran  E  ->  E! w  e.  { A ,  B }  { A ,  w }  e.  ran  E ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 176    \/ wo 357    /\ wa 358    /\ w3a 934   A.wal 1530   E.wex 1531    = wceq 1632    e. wcel 1696   E!weu 2156    =/= wne 2459   E.wrex 2557   E!wreu 2558   {cpr 3654   {ctp 3655   class class class wbr 4039   ran crn 4706   USGrph cusg 28227
This theorem is referenced by:  3vfriswmgra  28429
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528  ax-cnex 8809  ax-resscn 8810  ax-1cn 8811  ax-icn 8812  ax-addcl 8813  ax-addrcl 8814  ax-mulcl 8815  ax-mulrcl 8816  ax-mulcom 8817  ax-addass 8818  ax-mulass 8819  ax-distr 8820  ax-i2m1 8821  ax-1ne0 8822  ax-1rid 8823  ax-rnegex 8824  ax-rrecex 8825  ax-cnre 8826  ax-pre-lttri 8827  ax-pre-lttrn 8828  ax-pre-ltadd 8829  ax-pre-mulgt0 8830
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rmo 2564  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-1st 6138  df-2nd 6139  df-riota 6320  df-recs 6404  df-rdg 6439  df-1o 6495  df-oadd 6499  df-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-card 7588  df-cda 7810  df-pnf 8885  df-mnf 8886  df-xr 8887  df-ltxr 8888  df-le 8889  df-sub 9055  df-neg 9056  df-nn 9763  df-2 9820  df-n0 9982  df-z 10041  df-uz 10247  df-fz 10799  df-hash 11354  df-usgra 28229
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