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Theorem tz6.12f 5562
Description: Function value, using bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 30-Aug-1999.)
Hypothesis
Ref Expression
tz6.12f.1  |-  F/_ y F
Assertion
Ref Expression
tz6.12f  |-  ( (
<. A ,  y >.  e.  F  /\  E! y
<. A ,  y >.  e.  F )  ->  ( F `  A )  =  y )
Distinct variable group:    y, A
Allowed substitution hint:    F( y)

Proof of Theorem tz6.12f
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 opeq2 3813 . . . . 5  |-  ( z  =  y  ->  <. A , 
z >.  =  <. A , 
y >. )
21eleq1d 2362 . . . 4  |-  ( z  =  y  ->  ( <. A ,  z >.  e.  F  <->  <. A ,  y
>.  e.  F ) )
3 tz6.12f.1 . . . . . . 7  |-  F/_ y F
43nfel2 2444 . . . . . 6  |-  F/ y
<. A ,  z >.  e.  F
5 nfv 1609 . . . . . 6  |-  F/ z
<. A ,  y >.  e.  F
64, 5, 2cbveu 2176 . . . . 5  |-  ( E! z <. A ,  z
>.  e.  F  <->  E! y <. A ,  y >.  e.  F )
76a1i 10 . . . 4  |-  ( z  =  y  ->  ( E! z <. A ,  z
>.  e.  F  <->  E! y <. A ,  y >.  e.  F ) )
82, 7anbi12d 691 . . 3  |-  ( z  =  y  ->  (
( <. A ,  z
>.  e.  F  /\  E! z <. A ,  z
>.  e.  F )  <->  ( <. A ,  y >.  e.  F  /\  E! y <. A , 
y >.  e.  F ) ) )
9 eqeq2 2305 . . 3  |-  ( z  =  y  ->  (
( F `  A
)  =  z  <->  ( F `  A )  =  y ) )
108, 9imbi12d 311 . 2  |-  ( z  =  y  ->  (
( ( <. A , 
z >.  e.  F  /\  E! z <. A ,  z
>.  e.  F )  -> 
( F `  A
)  =  z )  <-> 
( ( <. A , 
y >.  e.  F  /\  E! y <. A ,  y
>.  e.  F )  -> 
( F `  A
)  =  y ) ) )
11 tz6.12 5561 . 2  |-  ( (
<. A ,  z >.  e.  F  /\  E! z
<. A ,  z >.  e.  F )  ->  ( F `  A )  =  z )
1210, 11chvarv 1966 1  |-  ( (
<. A ,  y >.  e.  F  /\  E! y
<. A ,  y >.  e.  F )  ->  ( F `  A )  =  y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   E!weu 2156   F/_wnfc 2419   <.cop 3656   ` cfv 5271
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-iota 5235  df-fv 5279
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