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

Theorem tz6.12i 5631
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 5622 . . . . 5  |-  ( F `
 A )  e. 
_V
2 neeq1 2529 . . . . . . . 8  |-  ( ( F `  A )  =  y  ->  (
( F `  A
)  =/=  (/)  <->  y  =/=  (/) ) )
3 tz6.12-2 5599 . . . . . . . . . . 11  |-  ( -.  E! y  A F y  ->  ( F `  A )  =  (/) )
43necon1ai 2563 . . . . . . . . . 10  |-  ( ( F `  A )  =/=  (/)  ->  E! y  A F y )
5 tz6.12c 5630 . . . . . . . . . 10  |-  ( E! y  A F y  ->  ( ( F `
 A )  =  y  <->  A F y ) )
64, 5syl 15 . . . . . . . . 9  |-  ( ( F `  A )  =/=  (/)  ->  ( ( F `  A )  =  y  <->  A F y ) )
76biimpcd 215 . . . . . . . 8  |-  ( ( F `  A )  =  y  ->  (
( F `  A
)  =/=  (/)  ->  A F y ) )
82, 7sylbird 226 . . . . . . 7  |-  ( ( F `  A )  =  y  ->  (
y  =/=  (/)  ->  A F y ) )
98eqcoms 2361 . . . . . 6  |-  ( y  =  ( F `  A )  ->  (
y  =/=  (/)  ->  A F y ) )
10 neeq1 2529 . . . . . 6  |-  ( y  =  ( F `  A )  ->  (
y  =/=  (/)  <->  ( F `  A )  =/=  (/) ) )
11 breq2 4108 . . . . . 6  |-  ( y  =  ( F `  A )  ->  ( A F y  <->  A F
( F `  A
) ) )
129, 10, 113imtr3d 258 . . . . 5  |-  ( y  =  ( F `  A )  ->  (
( F `  A
)  =/=  (/)  ->  A F ( F `  A ) ) )
131, 12vtocle 2933 . . . 4  |-  ( ( F `  A )  =/=  (/)  ->  A F
( F `  A
) )
1413a1i 10 . . 3  |-  ( ( F `  A )  =  B  ->  (
( F `  A
)  =/=  (/)  ->  A F ( F `  A ) ) )
15 neeq1 2529 . . 3  |-  ( ( F `  A )  =  B  ->  (
( F `  A
)  =/=  (/)  <->  B  =/=  (/) ) )
16 breq2 4108 . . 3  |-  ( ( F `  A )  =  B  ->  ( A F ( F `  A )  <->  A F B ) )
1714, 15, 163imtr3d 258 . 2  |-  ( ( F `  A )  =  B  ->  ( B  =/=  (/)  ->  A F B ) )
1817com12 27 1  |-  ( B  =/=  (/)  ->  ( ( F `  A )  =  B  ->  A F B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    = wceq 1642   E!weu 2209    =/= wne 2521   (/)c0 3531   class class class wbr 4104   ` cfv 5337
This theorem is referenced by:  fvbr0  5632  fvclss  5846  dcomex  8163
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-nul 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3909  df-br 4105  df-iota 5301  df-fv 5345
  Copyright terms: Public domain W3C validator