MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tz6.12i Unicode version

Theorem tz6.12i 5691
Description: Corollary of Theorem 6.12(2) of [TakeutiZaring] p. 27. (Contributed by Mario Carneiro, 17-Nov-2014.)
Assertion
Ref Expression
tz6.12i  |-  ( B  =/=  (/)  ->  ( ( F `  A )  =  B  ->  A F B ) )

Proof of Theorem tz6.12i
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 fvex 5682 . . . . 5  |-  ( F `
 A )  e. 
_V
2 neeq1 2558 . . . . . . . 8  |-  ( ( F `  A )  =  y  ->  (
( F `  A
)  =/=  (/)  <->  y  =/=  (/) ) )
3 tz6.12-2 5659 . . . . . . . . . . 11  |-  ( -.  E! y  A F y  ->  ( F `  A )  =  (/) )
43necon1ai 2592 . . . . . . . . . 10  |-  ( ( F `  A )  =/=  (/)  ->  E! y  A F y )
5 tz6.12c 5690 . . . . . . . . . 10  |-  ( E! y  A F y  ->  ( ( F `
 A )  =  y  <->  A F y ) )
64, 5syl 16 . . . . . . . . 9  |-  ( ( F `  A )  =/=  (/)  ->  ( ( F `  A )  =  y  <->  A F y ) )
76biimpcd 216 . . . . . . . 8  |-  ( ( F `  A )  =  y  ->  (
( F `  A
)  =/=  (/)  ->  A F y ) )
82, 7sylbird 227 . . . . . . 7  |-  ( ( F `  A )  =  y  ->  (
y  =/=  (/)  ->  A F y ) )
98eqcoms 2390 . . . . . 6  |-  ( y  =  ( F `  A )  ->  (
y  =/=  (/)  ->  A F y ) )
10 neeq1 2558 . . . . . 6  |-  ( y  =  ( F `  A )  ->  (
y  =/=  (/)  <->  ( F `  A )  =/=  (/) ) )
11 breq2 4157 . . . . . 6  |-  ( y  =  ( F `  A )  ->  ( A F y  <->  A F
( F `  A
) ) )
129, 10, 113imtr3d 259 . . . . 5  |-  ( y  =  ( F `  A )  ->  (
( F `  A
)  =/=  (/)  ->  A F ( F `  A ) ) )
131, 12vtocle 2968 . . . 4  |-  ( ( F `  A )  =/=  (/)  ->  A F
( F `  A
) )
1413a1i 11 . . 3  |-  ( ( F `  A )  =  B  ->  (
( F `  A
)  =/=  (/)  ->  A F ( F `  A ) ) )
15 neeq1 2558 . . 3  |-  ( ( F `  A )  =  B  ->  (
( F `  A
)  =/=  (/)  <->  B  =/=  (/) ) )
16 breq2 4157 . . 3  |-  ( ( F `  A )  =  B  ->  ( A F ( F `  A )  <->  A F B ) )
1714, 15, 163imtr3d 259 . 2  |-  ( ( F `  A )  =  B  ->  ( B  =/=  (/)  ->  A F B ) )
1817com12 29 1  |-  ( B  =/=  (/)  ->  ( ( F `  A )  =  B  ->  A F B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649   E!weu 2238    =/= wne 2550   (/)c0 3571   class class class wbr 4153   ` cfv 5394
This theorem is referenced by:  fvbr0  5692  fvclss  5919  dcomex  8260
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-nul 4279
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-iota 5358  df-fv 5402
  Copyright terms: Public domain W3C validator