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Theorem tz6.12f 5545
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 3780 . . . . 5  |-  ( z  =  y  ->  <. A , 
z >.  =  <. A , 
y >. )
21eleq1d 2246 . . . 4  |-  ( z  =  y  ->  ( <. A ,  z >.  e.  F  <->  <. A ,  y
>.  e.  F ) )
3 tz6.12f.1 . . . . . . 7  |-  F/_ y F
43nfel2 2332 . . . . . 6  |-  F/ y
<. A ,  z >.  e.  F
5 nfv 1528 . . . . . 6  |-  F/ z
<. A ,  y >.  e.  F
64, 5, 2cbveu 2050 . . . . 5  |-  ( E! z <. A ,  z
>.  e.  F  <->  E! y <. A ,  y >.  e.  F )
76a1i 9 . . . 4  |-  ( z  =  y  ->  ( E! z <. A ,  z
>.  e.  F  <->  E! y <. A ,  y >.  e.  F ) )
82, 7anbi12d 473 . . 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 2187 . . 3  |-  ( z  =  y  ->  (
( F `  A
)  =  z  <->  ( F `  A )  =  y ) )
108, 9imbi12d 234 . 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 5544 . 2  |-  ( (
<. A ,  z >.  e.  F  /\  E! z
<. A ,  z >.  e.  F )  ->  ( F `  A )  =  z )
1210, 11chvarv 1937 1  |-  ( (
<. A ,  y >.  e.  F  /\  E! y
<. A ,  y >.  e.  F )  ->  ( F `  A )  =  y )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    = wceq 1353   E!weu 2026    e. wcel 2148   F/_wnfc 2306   <.cop 3596   ` cfv 5217
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2740  df-sbc 2964  df-un 3134  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-br 4005  df-iota 5179  df-fv 5225
This theorem is referenced by: (None)
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