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Theorem nfixpxy 6747
Description: Bound-variable hypothesis builder for indexed Cartesian product. (Contributed by Mario Carneiro, 15-Oct-2016.) (Revised by Jim Kingdon, 15-Feb-2023.)
Hypotheses
Ref Expression
nfixp.1  |-  F/_ y A
nfixp.2  |-  F/_ y B
Assertion
Ref Expression
nfixpxy  |-  F/_ y X_ x  e.  A  B
Distinct variable group:    x, y
Allowed substitution hints:    A( x, y)    B( x, y)

Proof of Theorem nfixpxy
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 df-ixp 6729 . 2  |-  X_ x  e.  A  B  =  { z  |  ( z  Fn  { x  |  x  e.  A }  /\  A. x  e.  A  ( z `  x )  e.  B
) }
2 nfcv 2332 . . . . 5  |-  F/_ y
z
3 nftru 1477 . . . . . . 7  |-  F/ x T.
4 nfcvd 2333 . . . . . . . 8  |-  ( T. 
->  F/_ y x )
5 nfixp.1 . . . . . . . . 9  |-  F/_ y A
65a1i 9 . . . . . . . 8  |-  ( T. 
->  F/_ y A )
74, 6nfeld 2348 . . . . . . 7  |-  ( T. 
->  F/ y  x  e.  A )
83, 7nfabd 2352 . . . . . 6  |-  ( T. 
->  F/_ y { x  |  x  e.  A } )
98mptru 1373 . . . . 5  |-  F/_ y { x  |  x  e.  A }
102, 9nffn 5334 . . . 4  |-  F/ y  z  Fn  { x  |  x  e.  A }
11 df-ral 2473 . . . . 5  |-  ( A. x  e.  A  (
z `  x )  e.  B  <->  A. x ( x  e.  A  ->  (
z `  x )  e.  B ) )
122a1i 9 . . . . . . . . . 10  |-  ( T. 
->  F/_ y z )
1312, 4nffvd 5549 . . . . . . . . 9  |-  ( T. 
->  F/_ y ( z `
 x ) )
14 nfixp.2 . . . . . . . . . 10  |-  F/_ y B
1514a1i 9 . . . . . . . . 9  |-  ( T. 
->  F/_ y B )
1613, 15nfeld 2348 . . . . . . . 8  |-  ( T. 
->  F/ y ( z `
 x )  e.  B )
177, 16nfimd 1596 . . . . . . 7  |-  ( T. 
->  F/ y ( x  e.  A  ->  (
z `  x )  e.  B ) )
183, 17nfald 1771 . . . . . 6  |-  ( T. 
->  F/ y A. x
( x  e.  A  ->  ( z `  x
)  e.  B ) )
1918mptru 1373 . . . . 5  |-  F/ y A. x ( x  e.  A  ->  (
z `  x )  e.  B )
2011, 19nfxfr 1485 . . . 4  |-  F/ y A. x  e.  A  ( z `  x
)  e.  B
2110, 20nfan 1576 . . 3  |-  F/ y ( z  Fn  {
x  |  x  e.  A }  /\  A. x  e.  A  (
z `  x )  e.  B )
2221nfab 2337 . 2  |-  F/_ y { z  |  ( z  Fn  { x  |  x  e.  A }  /\  A. x  e.  A  ( z `  x )  e.  B
) }
231, 22nfcxfr 2329 1  |-  F/_ y X_ x  e.  A  B
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104   A.wal 1362   T. wtru 1365   F/wnf 1471    e. wcel 2160   {cab 2175   F/_wnfc 2319   A.wral 2468    Fn wfn 5233   ` cfv 5238   X_cixp 6728
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-br 4022  df-opab 4083  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-iota 5199  df-fun 5240  df-fn 5241  df-fv 5246  df-ixp 6729
This theorem is referenced by: (None)
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