MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  hashrabsn1 Structured version   Unicode version

Theorem hashrabsn1 12409
Description: If the size of a restricted class abstraction restricted to a singleton is 1, the condition of the class abstraction must hold for the singleton. (Contributed by Alexander van der Vekens, 3-Sep-2018.)
Assertion
Ref Expression
hashrabsn1  |-  ( (
# `  { x  e.  { A }  |  ph } )  =  1  ->  [. A  /  x ]. ph )
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem hashrabsn1
StepHypRef Expression
1 eqid 2467 . 2  |-  { x  e.  { A }  |  ph }  =  { x  e.  { A }  |  ph }
2 rabrsn 4097 . 2  |-  ( { x  e.  { A }  |  ph }  =  { x  e.  { A }  |  ph }  ->  ( { x  e.  { A }  |  ph }  =  (/)  \/  { x  e.  { A }  |  ph }  =  { A } ) )
3 fveq2 5865 . . . . 5  |-  ( { x  e.  { A }  |  ph }  =  (/) 
->  ( # `  {
x  e.  { A }  |  ph } )  =  ( # `  (/) ) )
43eqeq1d 2469 . . . 4  |-  ( { x  e.  { A }  |  ph }  =  (/) 
->  ( ( # `  {
x  e.  { A }  |  ph } )  =  1  <->  ( # `  (/) )  =  1 ) )
5 hash0 12404 . . . . . 6  |-  ( # `  (/) )  =  0
65eqeq1i 2474 . . . . 5  |-  ( (
# `  (/) )  =  1  <->  0  =  1 )
7 0ne1 10602 . . . . . 6  |-  0  =/=  1
8 eqneqall 2674 . . . . . 6  |-  ( 0  =  1  ->  (
0  =/=  1  ->  [. A  /  x ]. ph ) )
97, 8mpi 17 . . . . 5  |-  ( 0  =  1  ->  [. A  /  x ]. ph )
106, 9sylbi 195 . . . 4  |-  ( (
# `  (/) )  =  1  ->  [. A  /  x ]. ph )
114, 10syl6bi 228 . . 3  |-  ( { x  e.  { A }  |  ph }  =  (/) 
->  ( ( # `  {
x  e.  { A }  |  ph } )  =  1  ->  [. A  /  x ]. ph )
)
12 snidg 4053 . . . . . . . . 9  |-  ( A  e.  _V  ->  A  e.  { A } )
1312adantr 465 . . . . . . . 8  |-  ( ( A  e.  _V  /\  { x  e.  { A }  |  ph }  =  { A } )  ->  A  e.  { A } )
14 eleq2 2540 . . . . . . . . 9  |-  ( { x  e.  { A }  |  ph }  =  { A }  ->  ( A  e.  { x  e.  { A }  |  ph }  <->  A  e.  { A } ) )
1514adantl 466 . . . . . . . 8  |-  ( ( A  e.  _V  /\  { x  e.  { A }  |  ph }  =  { A } )  -> 
( A  e.  {
x  e.  { A }  |  ph }  <->  A  e.  { A } ) )
1613, 15mpbird 232 . . . . . . 7  |-  ( ( A  e.  _V  /\  { x  e.  { A }  |  ph }  =  { A } )  ->  A  e.  { x  e.  { A }  |  ph } )
17 nfcv 2629 . . . . . . . . 9  |-  F/_ x { A }
1817elrabsf 3370 . . . . . . . 8  |-  ( A  e.  { x  e. 
{ A }  |  ph }  <->  ( A  e. 
{ A }  /\  [. A  /  x ]. ph ) )
1918simprbi 464 . . . . . . 7  |-  ( A  e.  { x  e. 
{ A }  |  ph }  ->  [. A  /  x ]. ph )
2016, 19syl 16 . . . . . 6  |-  ( ( A  e.  _V  /\  { x  e.  { A }  |  ph }  =  { A } )  ->  [. A  /  x ]. ph )
2120a1d 25 . . . . 5  |-  ( ( A  e.  _V  /\  { x  e.  { A }  |  ph }  =  { A } )  -> 
( ( # `  {
x  e.  { A }  |  ph } )  =  1  ->  [. A  /  x ]. ph )
)
2221ex 434 . . . 4  |-  ( A  e.  _V  ->  ( { x  e.  { A }  |  ph }  =  { A }  ->  (
( # `  { x  e.  { A }  |  ph } )  =  1  ->  [. A  /  x ]. ph ) ) )
23 snprc 4091 . . . . 5  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
24 eqeq2 2482 . . . . . 6  |-  ( { A }  =  (/)  ->  ( { x  e. 
{ A }  |  ph }  =  { A } 
<->  { x  e.  { A }  |  ph }  =  (/) ) )
25 ax-1ne0 9560 . . . . . . . . . 10  |-  1  =/=  0
26 eqneqall 2674 . . . . . . . . . 10  |-  ( 1  =  0  ->  (
1  =/=  0  ->  [. A  /  x ]. ph ) )
2725, 26mpi 17 . . . . . . . . 9  |-  ( 1  =  0  ->  [. A  /  x ]. ph )
2827eqcoms 2479 . . . . . . . 8  |-  ( 0  =  1  ->  [. A  /  x ]. ph )
296, 28sylbi 195 . . . . . . 7  |-  ( (
# `  (/) )  =  1  ->  [. A  /  x ]. ph )
304, 29syl6bi 228 . . . . . 6  |-  ( { x  e.  { A }  |  ph }  =  (/) 
->  ( ( # `  {
x  e.  { A }  |  ph } )  =  1  ->  [. A  /  x ]. ph )
)
3124, 30syl6bi 228 . . . . 5  |-  ( { A }  =  (/)  ->  ( { x  e. 
{ A }  |  ph }  =  { A }  ->  ( ( # `  { x  e.  { A }  |  ph }
)  =  1  ->  [. A  /  x ]. ph ) ) )
3223, 31sylbi 195 . . . 4  |-  ( -.  A  e.  _V  ->  ( { x  e.  { A }  |  ph }  =  { A }  ->  ( ( # `  {
x  e.  { A }  |  ph } )  =  1  ->  [. A  /  x ]. ph )
) )
3322, 32pm2.61i 164 . . 3  |-  ( { x  e.  { A }  |  ph }  =  { A }  ->  (
( # `  { x  e.  { A }  |  ph } )  =  1  ->  [. A  /  x ]. ph ) )
3411, 33jaoi 379 . 2  |-  ( ( { x  e.  { A }  |  ph }  =  (/)  \/  { x  e.  { A }  |  ph }  =  { A } )  ->  (
( # `  { x  e.  { A }  |  ph } )  =  1  ->  [. A  /  x ]. ph ) )
351, 2, 34mp2b 10 1  |-  ( (
# `  { x  e.  { A }  |  ph } )  =  1  ->  [. A  /  x ]. ph )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1379    e. wcel 1767    =/= wne 2662   {crab 2818   _Vcvv 3113   [.wsbc 3331   (/)c0 3785   {csn 4027   ` cfv 5587   0cc0 9491   1c1 9492   #chash 12372
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6575  ax-cnex 9547  ax-resscn 9548  ax-1cn 9549  ax-icn 9550  ax-addcl 9551  ax-addrcl 9552  ax-mulcl 9553  ax-mulrcl 9554  ax-mulcom 9555  ax-addass 9556  ax-mulass 9557  ax-distr 9558  ax-i2m1 9559  ax-1ne0 9560  ax-1rid 9561  ax-rnegex 9562  ax-rrecex 9563  ax-cnre 9564  ax-pre-lttri 9565  ax-pre-lttrn 9566  ax-pre-ltadd 9567  ax-pre-mulgt0 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5550  df-fun 5589  df-fn 5590  df-f 5591  df-f1 5592  df-fo 5593  df-f1o 5594  df-fv 5595  df-riota 6244  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-om 6680  df-1st 6784  df-2nd 6785  df-recs 7042  df-rdg 7076  df-1o 7130  df-er 7311  df-en 7517  df-dom 7518  df-sdom 7519  df-fin 7520  df-card 8319  df-pnf 9629  df-mnf 9630  df-xr 9631  df-ltxr 9632  df-le 9633  df-sub 9806  df-neg 9807  df-nn 10536  df-n0 10795  df-z 10864  df-uz 11082  df-fz 11672  df-hash 12373
This theorem is referenced by:  rusgrasn  24637
  Copyright terms: Public domain W3C validator