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Theorem tz6.12-1 5710
Description: Function value. Theorem 6.12(1) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.)
Assertion
Ref Expression
tz6.12-1  |-  ( ( A F y  /\  E! y  A F
y )  ->  ( F `  A )  =  y )
Distinct variable groups:    y, F    y, A

Proof of Theorem tz6.12-1
StepHypRef Expression
1 df-fv 5425 . 2  |-  ( F `
 A )  =  ( iota y A F y )
2 iota1 5395 . . 3  |-  ( E! y  A F y  ->  ( A F y  <->  ( iota y A F y )  =  y ) )
32biimpac 473 . 2  |-  ( ( A F y  /\  E! y  A F
y )  ->  ( iota y A F y )  =  y )
41, 3syl5eq 2452 1  |-  ( ( A F y  /\  E! y  A F
y )  ->  ( F `  A )  =  y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    = wceq 1649   E!weu 2258   class class class wbr 4176   iotacio 5379   ` cfv 5417
This theorem is referenced by:  tz6.12  5711  tz6.12c  5713  funbrfv  5728
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 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-rex 2676  df-v 2922  df-sbc 3126  df-un 3289  df-sn 3784  df-pr 3785  df-uni 3980  df-iota 5381  df-fv 5425
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