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Theorem tz6.12f 5689
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 3927 . . . . 5  |-  ( z  =  y  ->  <. A , 
z >.  =  <. A , 
y >. )
21eleq1d 2453 . . . 4  |-  ( z  =  y  ->  ( <. A ,  z >.  e.  F  <->  <. A ,  y
>.  e.  F ) )
3 tz6.12f.1 . . . . . . 7  |-  F/_ y F
43nfel2 2535 . . . . . 6  |-  F/ y
<. A ,  z >.  e.  F
5 nfv 1626 . . . . . 6  |-  F/ z
<. A ,  y >.  e.  F
64, 5, 2cbveu 2258 . . . . 5  |-  ( E! z <. A ,  z
>.  e.  F  <->  E! y <. A ,  y >.  e.  F )
76a1i 11 . . . 4  |-  ( z  =  y  ->  ( E! z <. A ,  z
>.  e.  F  <->  E! y <. A ,  y >.  e.  F ) )
82, 7anbi12d 692 . . 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 2396 . . 3  |-  ( z  =  y  ->  (
( F `  A
)  =  z  <->  ( F `  A )  =  y ) )
108, 9imbi12d 312 . 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 5688 . 2  |-  ( (
<. A ,  z >.  e.  F  /\  E! z
<. A ,  z >.  e.  F )  ->  ( F `  A )  =  z )
1210, 11chvarv 2047 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 177    /\ wa 359    = wceq 1649    e. wcel 1717   E!weu 2238   F/_wnfc 2510   <.cop 3760   ` cfv 5394
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 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-iota 5358  df-fv 5402
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