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Theorem tz6.12i 5548
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 5539 . . . . 5  |-  ( F `
 A )  e. 
_V
2 neeq1 2454 . . . . . . . 8  |-  ( ( F `  A )  =  y  ->  (
( F `  A
)  =/=  (/)  <->  y  =/=  (/) ) )
3 tz6.12-2 5516 . . . . . . . . . . 11  |-  ( -.  E! y  A F y  ->  ( F `  A )  =  (/) )
43necon1ai 2488 . . . . . . . . . 10  |-  ( ( F `  A )  =/=  (/)  ->  E! y  A F y )
5 tz6.12c 5547 . . . . . . . . . 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 2286 . . . . . 6  |-  ( y  =  ( F `  A )  ->  (
y  =/=  (/)  ->  A F y ) )
10 neeq1 2454 . . . . . 6  |-  ( y  =  ( F `  A )  ->  (
y  =/=  (/)  <->  ( F `  A )  =/=  (/) ) )
11 breq2 4027 . . . . . 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 2857 . . . 4  |-  ( ( F `  A )  =/=  (/)  ->  A F
( F `  A
) )
1413a1i 10 . . 3  |-  ( ( F `  A )  =  B  ->  (
( F `  A
)  =/=  (/)  ->  A F ( F `  A ) ) )
15 neeq1 2454 . . 3  |-  ( ( F `  A )  =  B  ->  (
( F `  A
)  =/=  (/)  <->  B  =/=  (/) ) )
16 breq2 4027 . . 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 1623   E!weu 2143    =/= wne 2446   (/)c0 3455   class class class wbr 4023   ` cfv 5255
This theorem is referenced by:  fvbr0  5549  fvclss  5760  dcomex  8073
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-nul 4149
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263
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