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Theorem fsnex 6197
Description: Relate a function with a singleton as domain and one variable. (Contributed by Thierry Arnoux, 12-Jul-2020.)
Hypothesis
Ref Expression
fsnex.1  |-  ( x  =  ( f `  A )  ->  ( ps 
<-> 
ph ) )
Assertion
Ref Expression
fsnex  |-  ( A  e.  V  ->  ( E. f ( f : { A } --> D  /\  ph )  <->  E. x  e.  D  ps ) )
Distinct variable groups:    A, f, x    D, f, x    f, V, x    ps, f    ph, x
Allowed substitution hints:    ph( f)    ps( x)

Proof of Theorem fsnex
StepHypRef Expression
1 fsn2g 6080 . . . . . . . 8  |-  ( A  e.  V  ->  (
f : { A }
--> D  <->  ( ( f `
 A )  e.  D  /\  f  =  { <. A ,  ( f `  A )
>. } ) ) )
21simprbda 627 . . . . . . 7  |-  ( ( A  e.  V  /\  f : { A } --> D )  ->  (
f `  A )  e.  D )
32adantrr 721 . . . . . 6  |-  ( ( A  e.  V  /\  ( f : { A } --> D  /\  ph ) )  ->  (
f `  A )  e.  D )
4 fsnex.1 . . . . . . 7  |-  ( x  =  ( f `  A )  ->  ( ps 
<-> 
ph ) )
54adantl 467 . . . . . 6  |-  ( ( ( A  e.  V  /\  ( f : { A } --> D  /\  ph ) )  /\  x  =  ( f `  A ) )  -> 
( ps  <->  ph ) )
6 simprr 764 . . . . . 6  |-  ( ( A  e.  V  /\  ( f : { A } --> D  /\  ph ) )  ->  ph )
73, 5, 6rspcedvd 3187 . . . . 5  |-  ( ( A  e.  V  /\  ( f : { A } --> D  /\  ph ) )  ->  E. x  e.  D  ps )
87ex 435 . . . 4  |-  ( A  e.  V  ->  (
( f : { A } --> D  /\  ph )  ->  E. x  e.  D  ps ) )
98exlimdv 1772 . . 3  |-  ( A  e.  V  ->  ( E. f ( f : { A } --> D  /\  ph )  ->  E. x  e.  D  ps )
)
109imp 430 . 2  |-  ( ( A  e.  V  /\  E. f ( f : { A } --> D  /\  ph ) )  ->  E. x  e.  D  ps )
11 nfv 1755 . . . 4  |-  F/ x  A  e.  V
12 nfre1 2883 . . . 4  |-  F/ x E. x  e.  D  ps
1311, 12nfan 1988 . . 3  |-  F/ x
( A  e.  V  /\  E. x  e.  D  ps )
14 vex 3083 . . . . . . . . 9  |-  x  e. 
_V
15 f1osng 5870 . . . . . . . . 9  |-  ( ( A  e.  V  /\  x  e.  _V )  ->  { <. A ,  x >. } : { A }
-1-1-onto-> { x } )
1614, 15mpan2 675 . . . . . . . 8  |-  ( A  e.  V  ->  { <. A ,  x >. } : { A } -1-1-onto-> { x } )
1716ad3antrrr 734 . . . . . . 7  |-  ( ( ( ( A  e.  V  /\  E. x  e.  D  ps )  /\  x  e.  D
)  /\  ps )  ->  { <. A ,  x >. } : { A }
-1-1-onto-> { x } )
18 f1of 5831 . . . . . . 7  |-  ( {
<. A ,  x >. } : { A } -1-1-onto-> {
x }  ->  { <. A ,  x >. } : { A } --> { x } )
1917, 18syl 17 . . . . . 6  |-  ( ( ( ( A  e.  V  /\  E. x  e.  D  ps )  /\  x  e.  D
)  /\  ps )  ->  { <. A ,  x >. } : { A }
--> { x } )
20 simplr 760 . . . . . . 7  |-  ( ( ( ( A  e.  V  /\  E. x  e.  D  ps )  /\  x  e.  D
)  /\  ps )  ->  x  e.  D )
2120snssd 4145 . . . . . 6  |-  ( ( ( ( A  e.  V  /\  E. x  e.  D  ps )  /\  x  e.  D
)  /\  ps )  ->  { x }  C_  D )
22 fss 5754 . . . . . 6  |-  ( ( { <. A ,  x >. } : { A }
--> { x }  /\  { x }  C_  D
)  ->  { <. A ,  x >. } : { A } --> D )
2319, 21, 22syl2anc 665 . . . . 5  |-  ( ( ( ( A  e.  V  /\  E. x  e.  D  ps )  /\  x  e.  D
)  /\  ps )  ->  { <. A ,  x >. } : { A }
--> D )
24 fvsng 6114 . . . . . . . 8  |-  ( ( A  e.  V  /\  x  e.  _V )  ->  ( { <. A ,  x >. } `  A
)  =  x )
2514, 24mpan2 675 . . . . . . 7  |-  ( A  e.  V  ->  ( { <. A ,  x >. } `  A )  =  x )
2625eqcomd 2430 . . . . . 6  |-  ( A  e.  V  ->  x  =  ( { <. A ,  x >. } `  A ) )
2726ad3antrrr 734 . . . . 5  |-  ( ( ( ( A  e.  V  /\  E. x  e.  D  ps )  /\  x  e.  D
)  /\  ps )  ->  x  =  ( {
<. A ,  x >. } `
 A ) )
28 snex 4662 . . . . . 6  |-  { <. A ,  x >. }  e.  _V
29 feq1 5728 . . . . . . 7  |-  ( f  =  { <. A ,  x >. }  ->  (
f : { A }
--> D  <->  { <. A ,  x >. } : { A }
--> D ) )
30 fveq1 5881 . . . . . . . 8  |-  ( f  =  { <. A ,  x >. }  ->  (
f `  A )  =  ( { <. A ,  x >. } `  A ) )
3130eqeq2d 2436 . . . . . . 7  |-  ( f  =  { <. A ,  x >. }  ->  (
x  =  ( f `
 A )  <->  x  =  ( { <. A ,  x >. } `  A ) ) )
3229, 31anbi12d 715 . . . . . 6  |-  ( f  =  { <. A ,  x >. }  ->  (
( f : { A } --> D  /\  x  =  ( f `  A ) )  <->  ( { <. A ,  x >. } : { A } --> D  /\  x  =  ( { <. A ,  x >. } `  A ) ) ) )
3328, 32spcev 3173 . . . . 5  |-  ( ( { <. A ,  x >. } : { A }
--> D  /\  x  =  ( { <. A ,  x >. } `  A
) )  ->  E. f
( f : { A } --> D  /\  x  =  ( f `  A ) ) )
3423, 27, 33syl2anc 665 . . . 4  |-  ( ( ( ( A  e.  V  /\  E. x  e.  D  ps )  /\  x  e.  D
)  /\  ps )  ->  E. f ( f : { A } --> D  /\  x  =  ( f `  A ) ) )
35 simprl 762 . . . . . . 7  |-  ( ( ( ( ( A  e.  V  /\  E. x  e.  D  ps )  /\  x  e.  D
)  /\  ps )  /\  ( f : { A } --> D  /\  x  =  ( f `  A ) ) )  ->  f : { A } --> D )
36 simpllr 767 . . . . . . . . 9  |-  ( ( ( ( ( ( A  e.  V  /\  E. x  e.  D  ps )  /\  x  e.  D
)  /\  ps )  /\  ( f : { A } --> D  /\  x  =  ( f `  A ) ) )  /\  f : { A } --> D )  ->  ps )
37 simplrr 769 . . . . . . . . . 10  |-  ( ( ( ( ( ( A  e.  V  /\  E. x  e.  D  ps )  /\  x  e.  D
)  /\  ps )  /\  ( f : { A } --> D  /\  x  =  ( f `  A ) ) )  /\  f : { A } --> D )  ->  x  =  ( f `  A ) )
3837, 4syl 17 . . . . . . . . 9  |-  ( ( ( ( ( ( A  e.  V  /\  E. x  e.  D  ps )  /\  x  e.  D
)  /\  ps )  /\  ( f : { A } --> D  /\  x  =  ( f `  A ) ) )  /\  f : { A } --> D )  -> 
( ps  <->  ph ) )
3936, 38mpbid 213 . . . . . . . 8  |-  ( ( ( ( ( ( A  e.  V  /\  E. x  e.  D  ps )  /\  x  e.  D
)  /\  ps )  /\  ( f : { A } --> D  /\  x  =  ( f `  A ) ) )  /\  f : { A } --> D )  ->  ph )
4035, 39mpdan 672 . . . . . . 7  |-  ( ( ( ( ( A  e.  V  /\  E. x  e.  D  ps )  /\  x  e.  D
)  /\  ps )  /\  ( f : { A } --> D  /\  x  =  ( f `  A ) ) )  ->  ph )
4135, 40jca 534 . . . . . 6  |-  ( ( ( ( ( A  e.  V  /\  E. x  e.  D  ps )  /\  x  e.  D
)  /\  ps )  /\  ( f : { A } --> D  /\  x  =  ( f `  A ) ) )  ->  ( f : { A } --> D  /\  ph ) )
4241ex 435 . . . . 5  |-  ( ( ( ( A  e.  V  /\  E. x  e.  D  ps )  /\  x  e.  D
)  /\  ps )  ->  ( ( f : { A } --> D  /\  x  =  ( f `  A ) )  -> 
( f : { A } --> D  /\  ph ) ) )
4342eximdv 1758 . . . 4  |-  ( ( ( ( A  e.  V  /\  E. x  e.  D  ps )  /\  x  e.  D
)  /\  ps )  ->  ( E. f ( f : { A }
--> D  /\  x  =  ( f `  A
) )  ->  E. f
( f : { A } --> D  /\  ph ) ) )
4434, 43mpd 15 . . 3  |-  ( ( ( ( A  e.  V  /\  E. x  e.  D  ps )  /\  x  e.  D
)  /\  ps )  ->  E. f ( f : { A } --> D  /\  ph ) )
45 simpr 462 . . 3  |-  ( ( A  e.  V  /\  E. x  e.  D  ps )  ->  E. x  e.  D  ps )
4613, 44, 45r19.29af 2965 . 2  |-  ( ( A  e.  V  /\  E. x  e.  D  ps )  ->  E. f ( f : { A } --> D  /\  ph ) )
4710, 46impbida 840 1  |-  ( A  e.  V  ->  ( E. f ( f : { A } --> D  /\  ph )  <->  E. x  e.  D  ps ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437   E.wex 1657    e. wcel 1872   E.wrex 2772   _Vcvv 3080    C_ wss 3436   {csn 3998   <.cop 4004   -->wf 5597   -1-1-onto->wf1o 5600   ` cfv 5601
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546  ax-nul 4555  ax-pr 4660
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2273  df-mo 2274  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-ne 2616  df-ral 2776  df-rex 2777  df-reu 2778  df-rab 2780  df-v 3082  df-sbc 3300  df-dif 3439  df-un 3441  df-in 3443  df-ss 3450  df-nul 3762  df-if 3912  df-sn 3999  df-pr 4001  df-op 4005  df-uni 4220  df-br 4424  df-opab 4483  df-id 4768  df-xp 4859  df-rel 4860  df-cnv 4861  df-co 4862  df-dm 4863  df-rn 4864  df-res 4865  df-ima 4866  df-iota 5565  df-fun 5603  df-fn 5604  df-f 5605  df-f1 5606  df-fo 5607  df-f1o 5608  df-fv 5609
This theorem is referenced by: (None)
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