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Theorem fundmge2nop 39036
Description: A function with a domain containing (at least) two different elements is not an ordered pair. (Contributed by AV, 12-Oct-2020.)
Assertion
Ref Expression
fundmge2nop  |-  ( ( Fun  G  /\  2  <_  ( # `  dom  G ) )  ->  -.  G  e.  ( _V  X.  _V ) )

Proof of Theorem fundmge2nop
Dummy variables  a 
b  c  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dmexg 6729 . . . . . 6  |-  ( G  e.  _V  ->  dom  G  e.  _V )
2 hashge2el2dif 12644 . . . . . . 7  |-  ( ( dom  G  e.  _V  /\  2  <_  ( # `  dom  G ) )  ->  E. a  e.  dom  G E. b  e.  dom  G  a  =/=  b )
32ex 436 . . . . . 6  |-  ( dom 
G  e.  _V  ->  ( 2  <_  ( # `  dom  G )  ->  E. a  e.  dom  G E. b  e.  dom  G  a  =/=  b ) )
41, 3syl 17 . . . . 5  |-  ( G  e.  _V  ->  (
2  <_  ( # `  dom  G )  ->  E. a  e.  dom  G E. b  e.  dom  G  a  =/=  b ) )
5 df-ne 2626 . . . . . . . 8  |-  ( a  =/=  b  <->  -.  a  =  b )
6 elvv 4896 . . . . . . . . . . . 12  |-  ( G  e.  ( _V  X.  _V )  <->  E. x E. y  G  =  <. x ,  y >. )
7 funeq 5604 . . . . . . . . . . . . . . . . 17  |-  ( G  =  <. x ,  y
>.  ->  ( Fun  G  <->  Fun 
<. x ,  y >.
) )
8 vex 3050 . . . . . . . . . . . . . . . . . . 19  |-  x  e. 
_V
9 vex 3050 . . . . . . . . . . . . . . . . . . 19  |-  y  e. 
_V
108, 9funop 39029 . . . . . . . . . . . . . . . . . 18  |-  ( Fun 
<. x ,  y >.  <->  E. c ( x  =  { c }  /\  <.
x ,  y >.  =  { <. c ,  c
>. } ) )
11 eqeq2 2464 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( <.
x ,  y >.  =  { <. c ,  c
>. }  ->  ( G  =  <. x ,  y
>. 
<->  G  =  { <. c ,  c >. } ) )
12 dmeq 5038 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( G  =  { <. c ,  c >. }  ->  dom 
G  =  dom  { <. c ,  c >. } )
13 vex 3050 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  c  e. 
_V
1413dmsnop 5313 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  dom  { <. c ,  c >. }  =  { c }
1512, 14syl6eq 2503 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( G  =  { <. c ,  c >. }  ->  dom 
G  =  { c } )
16 eleq2 2520 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( dom 
G  =  { c }  ->  ( a  e.  dom  G  <->  a  e.  { c } ) )
17 elsn 3984 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( a  e.  { c }  <-> 
a  =  c )
1816, 17syl6bb 265 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( dom 
G  =  { c }  ->  ( a  e.  dom  G  <->  a  =  c ) )
19 eleq2 2520 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( dom 
G  =  { c }  ->  ( b  e.  dom  G  <->  b  e.  { c } ) )
20 elsn 3984 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( b  e.  { c }  <-> 
b  =  c )
2119, 20syl6bb 265 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( dom 
G  =  { c }  ->  ( b  e.  dom  G  <->  b  =  c ) )
22 equtr2 1871 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( a  =  c  /\  b  =  c )  ->  a  =  b )
2322a1d 26 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( a  =  c  /\  b  =  c )  ->  ( G  e.  _V  ->  a  =  b ) )
2423expcom 437 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( b  =  c  ->  (
a  =  c  -> 
( G  e.  _V  ->  a  =  b ) ) )
2521, 24syl6bi 232 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( dom 
G  =  { c }  ->  ( b  e.  dom  G  ->  (
a  =  c  -> 
( G  e.  _V  ->  a  =  b ) ) ) )
2625com23 81 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( dom 
G  =  { c }  ->  ( a  =  c  ->  ( b  e.  dom  G  -> 
( G  e.  _V  ->  a  =  b ) ) ) )
2718, 26sylbid 219 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( dom 
G  =  { c }  ->  ( a  e.  dom  G  ->  (
b  e.  dom  G  ->  ( G  e.  _V  ->  a  =  b ) ) ) )
2827impd 433 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( dom 
G  =  { c }  ->  ( (
a  e.  dom  G  /\  b  e.  dom  G )  ->  ( G  e.  _V  ->  a  =  b ) ) )
2915, 28syl 17 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( G  =  { <. c ,  c >. }  ->  ( ( a  e.  dom  G  /\  b  e.  dom  G )  ->  ( G  e.  _V  ->  a  =  b ) ) )
3011, 29syl6bi 232 . . . . . . . . . . . . . . . . . . . . 21  |-  ( <.
x ,  y >.  =  { <. c ,  c
>. }  ->  ( G  =  <. x ,  y
>.  ->  ( ( a  e.  dom  G  /\  b  e.  dom  G )  ->  ( G  e. 
_V  ->  a  =  b ) ) ) )
3130adantl 468 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( x  =  { c }  /\  <. x ,  y >.  =  { <. c ,  c >. } )  ->  ( G  =  <. x ,  y >.  ->  ( ( a  e.  dom  G  /\  b  e.  dom  G )  ->  ( G  e.  _V  ->  a  =  b ) ) ) )
3231exlimiv 1778 . . . . . . . . . . . . . . . . . . 19  |-  ( E. c ( x  =  { c }  /\  <.
x ,  y >.  =  { <. c ,  c
>. } )  ->  ( G  =  <. x ,  y >.  ->  ( ( a  e.  dom  G  /\  b  e.  dom  G )  ->  ( G  e.  _V  ->  a  =  b ) ) ) )
3332com12 32 . . . . . . . . . . . . . . . . . 18  |-  ( G  =  <. x ,  y
>.  ->  ( E. c
( x  =  {
c }  /\  <. x ,  y >.  =  { <. c ,  c >. } )  ->  (
( a  e.  dom  G  /\  b  e.  dom  G )  ->  ( G  e.  _V  ->  a  =  b ) ) ) )
3410, 33syl5bi 221 . . . . . . . . . . . . . . . . 17  |-  ( G  =  <. x ,  y
>.  ->  ( Fun  <. x ,  y >.  ->  (
( a  e.  dom  G  /\  b  e.  dom  G )  ->  ( G  e.  _V  ->  a  =  b ) ) ) )
357, 34sylbid 219 . . . . . . . . . . . . . . . 16  |-  ( G  =  <. x ,  y
>.  ->  ( Fun  G  ->  ( ( a  e. 
dom  G  /\  b  e.  dom  G )  -> 
( G  e.  _V  ->  a  =  b ) ) ) )
3635com23 81 . . . . . . . . . . . . . . 15  |-  ( G  =  <. x ,  y
>.  ->  ( ( a  e.  dom  G  /\  b  e.  dom  G )  ->  ( Fun  G  ->  ( G  e.  _V  ->  a  =  b ) ) ) )
37363impd 1224 . . . . . . . . . . . . . 14  |-  ( G  =  <. x ,  y
>.  ->  ( ( ( a  e.  dom  G  /\  b  e.  dom  G )  /\  Fun  G  /\  G  e.  _V )  ->  a  =  b ) )
3837exlimivv 1780 . . . . . . . . . . . . 13  |-  ( E. x E. y  G  =  <. x ,  y
>.  ->  ( ( ( a  e.  dom  G  /\  b  e.  dom  G )  /\  Fun  G  /\  G  e.  _V )  ->  a  =  b ) )
3938com12 32 . . . . . . . . . . . 12  |-  ( ( ( a  e.  dom  G  /\  b  e.  dom  G )  /\  Fun  G  /\  G  e.  _V )  ->  ( E. x E. y  G  =  <. x ,  y >.  ->  a  =  b ) )
406, 39syl5bi 221 . . . . . . . . . . 11  |-  ( ( ( a  e.  dom  G  /\  b  e.  dom  G )  /\  Fun  G  /\  G  e.  _V )  ->  ( G  e.  ( _V  X.  _V )  ->  a  =  b ) )
4140con3d 139 . . . . . . . . . 10  |-  ( ( ( a  e.  dom  G  /\  b  e.  dom  G )  /\  Fun  G  /\  G  e.  _V )  ->  ( -.  a  =  b  ->  -.  G  e.  ( _V  X.  _V ) ) )
42413exp 1208 . . . . . . . . 9  |-  ( ( a  e.  dom  G  /\  b  e.  dom  G )  ->  ( Fun  G  ->  ( G  e. 
_V  ->  ( -.  a  =  b  ->  -.  G  e.  ( _V  X.  _V ) ) ) ) )
4342com24 90 . . . . . . . 8  |-  ( ( a  e.  dom  G  /\  b  e.  dom  G )  ->  ( -.  a  =  b  ->  ( G  e.  _V  ->  ( Fun  G  ->  -.  G  e.  ( _V  X.  _V ) ) ) ) )
445, 43syl5bi 221 . . . . . . 7  |-  ( ( a  e.  dom  G  /\  b  e.  dom  G )  ->  ( a  =/=  b  ->  ( G  e.  _V  ->  ( Fun  G  ->  -.  G  e.  ( _V  X.  _V ) ) ) ) )
4544rexlimivv 2886 . . . . . 6  |-  ( E. a  e.  dom  G E. b  e.  dom  G  a  =/=  b  -> 
( G  e.  _V  ->  ( Fun  G  ->  -.  G  e.  ( _V  X.  _V ) ) ) )
4645com12 32 . . . . 5  |-  ( G  e.  _V  ->  ( E. a  e.  dom  G E. b  e.  dom  G  a  =/=  b  -> 
( Fun  G  ->  -.  G  e.  ( _V 
X.  _V ) ) ) )
474, 46syld 45 . . . 4  |-  ( G  e.  _V  ->  (
2  <_  ( # `  dom  G )  ->  ( Fun  G  ->  -.  G  e.  ( _V  X.  _V )
) ) )
4847com13 83 . . 3  |-  ( Fun 
G  ->  ( 2  <_  ( # `  dom  G )  ->  ( G  e.  _V  ->  -.  G  e.  ( _V  X.  _V ) ) ) )
4948imp 431 . 2  |-  ( ( Fun  G  /\  2  <_  ( # `  dom  G ) )  ->  ( G  e.  _V  ->  -.  G  e.  ( _V 
X.  _V ) ) )
50 elex 3056 . . 3  |-  ( G  e.  ( _V  X.  _V )  ->  G  e. 
_V )
5150con3i 141 . 2  |-  ( -.  G  e.  _V  ->  -.  G  e.  ( _V 
X.  _V ) )
5249, 51pm2.61d1 163 1  |-  ( ( Fun  G  /\  2  <_  ( # `  dom  G ) )  ->  -.  G  e.  ( _V  X.  _V ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 371    /\ w3a 986    = wceq 1446   E.wex 1665    e. wcel 1889    =/= wne 2624   E.wrex 2740   _Vcvv 3047   {csn 3970   <.cop 3976   class class class wbr 4405    X. cxp 4835   dom cdm 4837   Fun wfun 5579   ` cfv 5585    <_ cle 9681   2c2 10666   #chash 12522
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-fal 1452  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-pss 3422  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-tp 3975  df-op 3977  df-uni 4202  df-int 4238  df-iun 4283  df-br 4406  df-opab 4465  df-mpt 4466  df-tr 4501  df-eprel 4748  df-id 4752  df-po 4758  df-so 4759  df-fr 4796  df-we 4798  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-pred 5383  df-ord 5429  df-on 5430  df-lim 5431  df-suc 5432  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6698  df-1st 6798  df-2nd 6799  df-wrecs 7033  df-recs 7095  df-rdg 7133  df-1o 7187  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-fin 7578  df-card 8378  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868  df-nn 10617  df-2 10675  df-n0 10877  df-z 10945  df-uz 11167  df-fz 11792  df-hash 12523
This theorem is referenced by:  fun2dmnop  39037  funvtxdmge2val  39126  funiedgdmge2val  39127
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