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Theorem tz6.12i 5401
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 5391 . . . . 5  |-  ( F `
 A )  e. 
_V
2 neeq1 2420 . . . . . . . 8  |-  ( ( F `  A )  =  y  ->  (
( F `  A
)  =/=  (/)  <->  y  =/=  (/) ) )
3 tz6.12-2 5399 . . . . . . . . . . 11  |-  ( -.  E! y  A F y  ->  ( F `  A )  =  (/) )
43necon1ai 2454 . . . . . . . . . 10  |-  ( ( F `  A )  =/=  (/)  ->  E! y  A F y )
5 tz6.12c 5400 . . . . . . . . . 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 2256 . . . . . 6  |-  ( y  =  ( F `  A )  ->  (
y  =/=  (/)  ->  A F y ) )
10 neeq1 2420 . . . . . 6  |-  ( y  =  ( F `  A )  ->  (
y  =/=  (/)  <->  ( F `  A )  =/=  (/) ) )
11 breq2 3924 . . . . . 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 2795 . . . 4  |-  ( ( F `  A )  =/=  (/)  ->  A F
( F `  A
) )
1413a1i 12 . . 3  |-  ( ( F `  A )  =  B  ->  (
( F `  A
)  =/=  (/)  ->  A F ( F `  A ) ) )
15 neeq1 2420 . . 3  |-  ( ( F `  A )  =  B  ->  (
( F `  A
)  =/=  (/)  <->  B  =/=  (/) ) )
16 breq2 3924 . . 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 2114    =/= wne 2412   (/)c0 3362   class class class wbr 3920   ` cfv 4592
This theorem is referenced by:  fvbr0  5402  fvclss  5612  dcomex  7957
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 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pr 4108  ax-un 4403
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-xp 4594  df-cnv 4596  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fv 4608
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