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Theorem tz6.12f 5546
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 3797 . . . . 5  |-  ( z  =  y  ->  <. A , 
z >.  =  <. A , 
y >. )
21eleq1d 2349 . . . 4  |-  ( z  =  y  ->  ( <. A ,  z >.  e.  F  <->  <. A ,  y
>.  e.  F ) )
3 tz6.12f.1 . . . . . . 7  |-  F/_ y F
43nfel2 2431 . . . . . 6  |-  F/ y
<. A ,  z >.  e.  F
5 nfv 1605 . . . . . 6  |-  F/ z
<. A ,  y >.  e.  F
64, 5, 2cbveu 2163 . . . . 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 2292 . . 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 5545 . 2  |-  ( (
<. A ,  z >.  e.  F  /\  E! z
<. A ,  z >.  e.  F )  ->  ( F `  A )  =  z )
1210, 11chvarv 1953 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 1623    e. wcel 1684   E!weu 2143   F/_wnfc 2406   <.cop 3643   ` cfv 5255
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263
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