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Theorem tz6.12i 5482
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
StepHypRef Expression
1 fvex 5472 . . . . 5  |-  ( F `
 A )  e. 
_V
2 neeq1 2429 . . . . . . . 8  |-  ( ( F `  A )  =  y  ->  (
( F `  A
)  =/=  (/)  <->  y  =/=  (/) ) )
3 tz6.12-2 5480 . . . . . . . . . . 11  |-  ( -.  E! y  A F y  ->  ( F `  A )  =  (/) )
43necon1ai 2463 . . . . . . . . . 10  |-  ( ( F `  A )  =/=  (/)  ->  E! y  A F y )
5 tz6.12c 5481 . . . . . . . . . 10  |-  ( E! y  A F y  ->  ( ( F `
 A )  =  y  <->  A F y ) )
64, 5syl 17 . . . . . . . . 9  |-  ( ( F `  A )  =/=  (/)  ->  ( ( F `  A )  =  y  <->  A F y ) )
76biimpcd 217 . . . . . . . 8  |-  ( ( F `  A )  =  y  ->  (
( F `  A
)  =/=  (/)  ->  A F y ) )
82, 7sylbird 228 . . . . . . 7  |-  ( ( F `  A )  =  y  ->  (
y  =/=  (/)  ->  A F y ) )
98eqcoms 2261 . . . . . 6  |-  ( y  =  ( F `  A )  ->  (
y  =/=  (/)  ->  A F y ) )
10 neeq1 2429 . . . . . 6  |-  ( y  =  ( F `  A )  ->  (
y  =/=  (/)  <->  ( F `  A )  =/=  (/) ) )
11 breq2 4001 . . . . . 6  |-  ( y  =  ( F `  A )  ->  ( A F y  <->  A F
( F `  A
) ) )
129, 10, 113imtr3d 260 . . . . 5  |-  ( y  =  ( F `  A )  ->  (
( F `  A
)  =/=  (/)  ->  A F ( F `  A ) ) )
131, 12vtocle 2832 . . . 4  |-  ( ( F `  A )  =/=  (/)  ->  A F
( F `  A
) )
1413a1i 12 . . 3  |-  ( ( F `  A )  =  B  ->  (
( F `  A
)  =/=  (/)  ->  A F ( F `  A ) ) )
15 neeq1 2429 . . 3  |-  ( ( F `  A )  =  B  ->  (
( F `  A
)  =/=  (/)  <->  B  =/=  (/) ) )
16 breq2 4001 . . 3  |-  ( ( F `  A )  =  B  ->  ( A F ( F `  A )  <->  A F B ) )
1714, 15, 163imtr3d 260 . 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 6    <-> wb 178    = wceq 1619   E!weu 2118    =/= wne 2421   (/)c0 3430   class class class wbr 3997   ` cfv 4673
This theorem is referenced by:  fvbr0  5483  fvclss  5694  dcomex  8041
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186  ax-un 4484
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-sbc 2967  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-xp 4675  df-cnv 4677  df-dm 4679  df-rn 4680  df-res 4681  df-ima 4682  df-fv 4689
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