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Theorem fundmge2nop 39172
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 6743 . . . . . 6  |-  ( G  e.  _V  ->  dom  G  e.  _V )
2 hashge2el2dif 12678 . . . . . . 7  |-  ( ( dom  G  e.  _V  /\  2  <_  ( # `  dom  G ) )  ->  E. a  e.  dom  G E. b  e.  dom  G  a  =/=  b )
32ex 441 . . . . . 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 2643 . . . . . . . 8  |-  ( a  =/=  b  <->  -.  a  =  b )
6 elvv 4898 . . . . . . . . . . . 12  |-  ( G  e.  ( _V  X.  _V )  <->  E. x E. y  G  =  <. x ,  y >. )
7 funeq 5608 . . . . . . . . . . . . . . . . 17  |-  ( G  =  <. x ,  y
>.  ->  ( Fun  G  <->  Fun 
<. x ,  y >.
) )
8 vex 3034 . . . . . . . . . . . . . . . . . . 19  |-  x  e. 
_V
9 vex 3034 . . . . . . . . . . . . . . . . . . 19  |-  y  e. 
_V
108, 9funop 39165 . . . . . . . . . . . . . . . . . 18  |-  ( Fun 
<. x ,  y >.  <->  E. c ( x  =  { c }  /\  <.
x ,  y >.  =  { <. c ,  c
>. } ) )
11 eqeq2 2482 . . . . . . . . . . . . . . . . . . . . . 22  |-  ( <.
x ,  y >.  =  { <. c ,  c
>. }  ->  ( G  =  <. x ,  y
>. 
<->  G  =  { <. c ,  c >. } ) )
12 dmeq 5040 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( G  =  { <. c ,  c >. }  ->  dom 
G  =  dom  { <. c ,  c >. } )
13 vex 3034 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  c  e. 
_V
1413dmsnop 5317 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  dom  { <. c ,  c >. }  =  { c }
1512, 14syl6eq 2521 . . . . . . . . . . . . . . . . . . . . . . 23  |-  ( G  =  { <. c ,  c >. }  ->  dom 
G  =  { c } )
16 eleq2 2538 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( dom 
G  =  { c }  ->  ( a  e.  dom  G  <->  a  e.  { c } ) )
17 elsn 3973 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( a  e.  { c }  <-> 
a  =  c )
1816, 17syl6bb 269 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( dom 
G  =  { c }  ->  ( a  e.  dom  G  <->  a  =  c ) )
19 eleq2 2538 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( dom 
G  =  { c }  ->  ( b  e.  dom  G  <->  b  e.  { c } ) )
20 elsn 3973 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( b  e.  { c }  <-> 
b  =  c )
2119, 20syl6bb 269 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( dom 
G  =  { c }  ->  ( b  e.  dom  G  <->  b  =  c ) )
22 equtr2 1877 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29  |-  ( ( a  =  c  /\  b  =  c )  ->  a  =  b )
2322a1d 25 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28  |-  ( ( a  =  c  /\  b  =  c )  ->  ( G  e.  _V  ->  a  =  b ) )
2423expcom 442 . . . . . . . . . . . . . . . . . . . . . . . . . . 27  |-  ( b  =  c  ->  (
a  =  c  -> 
( G  e.  _V  ->  a  =  b ) ) )
2521, 24syl6bi 236 . . . . . . . . . . . . . . . . . . . . . . . . . 26  |-  ( dom 
G  =  { c }  ->  ( b  e.  dom  G  ->  (
a  =  c  -> 
( G  e.  _V  ->  a  =  b ) ) ) )
2625com23 80 . . . . . . . . . . . . . . . . . . . . . . . . 25  |-  ( dom 
G  =  { c }  ->  ( a  =  c  ->  ( b  e.  dom  G  -> 
( G  e.  _V  ->  a  =  b ) ) ) )
2718, 26sylbid 223 . . . . . . . . . . . . . . . . . . . . . . . 24  |-  ( dom 
G  =  { c }  ->  ( a  e.  dom  G  ->  (
b  e.  dom  G  ->  ( G  e.  _V  ->  a  =  b ) ) ) )
2827impd 438 . . . . . . . . . . . . . . . . . . . . . . 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 236 . . . . . . . . . . . . . . . . . . . . 21  |-  ( <.
x ,  y >.  =  { <. c ,  c
>. }  ->  ( G  =  <. x ,  y
>.  ->  ( ( a  e.  dom  G  /\  b  e.  dom  G )  ->  ( G  e. 
_V  ->  a  =  b ) ) ) )
3130adantl 473 . . . . . . . . . . . . . . . . . . . 20  |-  ( ( x  =  { c }  /\  <. x ,  y >.  =  { <. c ,  c >. } )  ->  ( G  =  <. x ,  y >.  ->  ( ( a  e.  dom  G  /\  b  e.  dom  G )  ->  ( G  e.  _V  ->  a  =  b ) ) ) )
3231exlimiv 1784 . . . . . . . . . . . . . . . . . . 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 31 . . . . . . . . . . . . . . . . . 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 225 . . . . . . . . . . . . . . . . 17  |-  ( G  =  <. x ,  y
>.  ->  ( Fun  <. x ,  y >.  ->  (
( a  e.  dom  G  /\  b  e.  dom  G )  ->  ( G  e.  _V  ->  a  =  b ) ) ) )
357, 34sylbid 223 . . . . . . . . . . . . . . . 16  |-  ( G  =  <. x ,  y
>.  ->  ( Fun  G  ->  ( ( a  e. 
dom  G  /\  b  e.  dom  G )  -> 
( G  e.  _V  ->  a  =  b ) ) ) )
3635com23 80 . . . . . . . . . . . . . . 15  |-  ( G  =  <. x ,  y
>.  ->  ( ( a  e.  dom  G  /\  b  e.  dom  G )  ->  ( Fun  G  ->  ( G  e.  _V  ->  a  =  b ) ) ) )
37363impd 1247 . . . . . . . . . . . . . 14  |-  ( G  =  <. x ,  y
>.  ->  ( ( ( a  e.  dom  G  /\  b  e.  dom  G )  /\  Fun  G  /\  G  e.  _V )  ->  a  =  b ) )
3837exlimivv 1786 . . . . . . . . . . . . 13  |-  ( E. x E. y  G  =  <. x ,  y
>.  ->  ( ( ( a  e.  dom  G  /\  b  e.  dom  G )  /\  Fun  G  /\  G  e.  _V )  ->  a  =  b ) )
3938com12 31 . . . . . . . . . . . 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 225 . . . . . . . . . . 11  |-  ( ( ( a  e.  dom  G  /\  b  e.  dom  G )  /\  Fun  G  /\  G  e.  _V )  ->  ( G  e.  ( _V  X.  _V )  ->  a  =  b ) )
4140con3d 140 . . . . . . . . . 10  |-  ( ( ( a  e.  dom  G  /\  b  e.  dom  G )  /\  Fun  G  /\  G  e.  _V )  ->  ( -.  a  =  b  ->  -.  G  e.  ( _V  X.  _V ) ) )
42413exp 1230 . . . . . . . . 9  |-  ( ( a  e.  dom  G  /\  b  e.  dom  G )  ->  ( Fun  G  ->  ( G  e. 
_V  ->  ( -.  a  =  b  ->  -.  G  e.  ( _V  X.  _V ) ) ) ) )
4342com24 89 . . . . . . . 8  |-  ( ( a  e.  dom  G  /\  b  e.  dom  G )  ->  ( -.  a  =  b  ->  ( G  e.  _V  ->  ( Fun  G  ->  -.  G  e.  ( _V  X.  _V ) ) ) ) )
445, 43syl5bi 225 . . . . . . 7  |-  ( ( a  e.  dom  G  /\  b  e.  dom  G )  ->  ( a  =/=  b  ->  ( G  e.  _V  ->  ( Fun  G  ->  -.  G  e.  ( _V  X.  _V ) ) ) ) )
4544rexlimivv 2876 . . . . . 6  |-  ( E. a  e.  dom  G E. b  e.  dom  G  a  =/=  b  -> 
( G  e.  _V  ->  ( Fun  G  ->  -.  G  e.  ( _V  X.  _V ) ) ) )
4645com12 31 . . . . 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 44 . . . 4  |-  ( G  e.  _V  ->  (
2  <_  ( # `  dom  G )  ->  ( Fun  G  ->  -.  G  e.  ( _V  X.  _V )
) ) )
4847com13 82 . . 3  |-  ( Fun 
G  ->  ( 2  <_  ( # `  dom  G )  ->  ( G  e.  _V  ->  -.  G  e.  ( _V  X.  _V ) ) ) )
4948imp 436 . 2  |-  ( ( Fun  G  /\  2  <_  ( # `  dom  G ) )  ->  ( G  e.  _V  ->  -.  G  e.  ( _V 
X.  _V ) ) )
50 elex 3040 . . 3  |-  ( G  e.  ( _V  X.  _V )  ->  G  e. 
_V )
5150con3i 142 . 2  |-  ( -.  G  e.  _V  ->  -.  G  e.  ( _V 
X.  _V ) )
5249, 51pm2.61d1 164 1  |-  ( ( Fun  G  /\  2  <_  ( # `  dom  G ) )  ->  -.  G  e.  ( _V  X.  _V ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 376    /\ w3a 1007    = wceq 1452   E.wex 1671    e. wcel 1904    =/= wne 2641   E.wrex 2757   _Vcvv 3031   {csn 3959   <.cop 3965   class class class wbr 4395    X. cxp 4837   dom cdm 4839   Fun wfun 5583   ` cfv 5589    <_ cle 9694   2c2 10681   #chash 12553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-fal 1458  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-1st 6812  df-2nd 6813  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-fin 7591  df-card 8391  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883  df-nn 10632  df-2 10690  df-n0 10894  df-z 10962  df-uz 11183  df-fz 11811  df-hash 12554
This theorem is referenced by:  fun2dmnop  39173  funvtxdmge2val  39269  funiedgdmge2val  39270
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