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Theorem hashrabsn1 12425
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 2454 . 2  |-  { x  e.  { A }  |  ph }  =  { x  e.  { A }  |  ph }
2 rabrsn 4086 . 2  |-  ( { x  e.  { A }  |  ph }  =  { x  e.  { A }  |  ph }  ->  ( { x  e.  { A }  |  ph }  =  (/)  \/  { x  e.  { A }  |  ph }  =  { A } ) )
3 fveq2 5848 . . . . 5  |-  ( { x  e.  { A }  |  ph }  =  (/) 
->  ( # `  {
x  e.  { A }  |  ph } )  =  ( # `  (/) ) )
43eqeq1d 2456 . . . 4  |-  ( { x  e.  { A }  |  ph }  =  (/) 
->  ( ( # `  {
x  e.  { A }  |  ph } )  =  1  <->  ( # `  (/) )  =  1 ) )
5 hash0 12420 . . . . . 6  |-  ( # `  (/) )  =  0
65eqeq1i 2461 . . . . 5  |-  ( (
# `  (/) )  =  1  <->  0  =  1 )
7 0ne1 10599 . . . . . 6  |-  0  =/=  1
8 eqneqall 2661 . . . . . 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 4042 . . . . . . . . 9  |-  ( A  e.  _V  ->  A  e.  { A } )
1312adantr 463 . . . . . . . 8  |-  ( ( A  e.  _V  /\  { x  e.  { A }  |  ph }  =  { A } )  ->  A  e.  { A } )
14 eleq2 2527 . . . . . . . . 9  |-  ( { x  e.  { A }  |  ph }  =  { A }  ->  ( A  e.  { x  e.  { A }  |  ph }  <->  A  e.  { A } ) )
1514adantl 464 . . . . . . . 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 2616 . . . . . . . . 9  |-  F/_ x { A }
1817elrabsf 3363 . . . . . . . 8  |-  ( A  e.  { x  e. 
{ A }  |  ph }  <->  ( A  e. 
{ A }  /\  [. A  /  x ]. ph ) )
1918simprbi 462 . . . . . . 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 432 . . . 4  |-  ( A  e.  _V  ->  ( { x  e.  { A }  |  ph }  =  { A }  ->  (
( # `  { x  e.  { A }  |  ph } )  =  1  ->  [. A  /  x ]. ph ) ) )
23 snprc 4079 . . . . 5  |-  ( -.  A  e.  _V  <->  { A }  =  (/) )
24 eqeq2 2469 . . . . . 6  |-  ( { A }  =  (/)  ->  ( { x  e. 
{ A }  |  ph }  =  { A } 
<->  { x  e.  { A }  |  ph }  =  (/) ) )
25 ax-1ne0 9550 . . . . . . . . . 10  |-  1  =/=  0
26 eqneqall 2661 . . . . . . . . . 10  |-  ( 1  =  0  ->  (
1  =/=  0  ->  [. A  /  x ]. ph ) )
2725, 26mpi 17 . . . . . . . . 9  |-  ( 1  =  0  ->  [. A  /  x ]. ph )
2827eqcoms 2466 . . . . . . . 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 377 . 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 366    /\ wa 367    = wceq 1398    e. wcel 1823    =/= wne 2649   {crab 2808   _Vcvv 3106   [.wsbc 3324   (/)c0 3783   {csn 4016   ` cfv 5570   0cc0 9481   1c1 9482   #chash 12387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565  ax-cnex 9537  ax-resscn 9538  ax-1cn 9539  ax-icn 9540  ax-addcl 9541  ax-addrcl 9542  ax-mulcl 9543  ax-mulrcl 9544  ax-mulcom 9545  ax-addass 9546  ax-mulass 9547  ax-distr 9548  ax-i2m1 9549  ax-1ne0 9550  ax-1rid 9551  ax-rnegex 9552  ax-rrecex 9553  ax-cnre 9554  ax-pre-lttri 9555  ax-pre-lttrn 9556  ax-pre-ltadd 9557  ax-pre-mulgt0 9558
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-int 4272  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-tr 4533  df-eprel 4780  df-id 4784  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871  df-lim 4872  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-riota 6232  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-om 6674  df-1st 6773  df-2nd 6774  df-recs 7034  df-rdg 7068  df-1o 7122  df-er 7303  df-en 7510  df-dom 7511  df-sdom 7512  df-fin 7513  df-card 8311  df-pnf 9619  df-mnf 9620  df-xr 9621  df-ltxr 9622  df-le 9623  df-sub 9798  df-neg 9799  df-nn 10532  df-n0 10792  df-z 10861  df-uz 11083  df-fz 11676  df-hash 12388
This theorem is referenced by:  rusgrasn  25147
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