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Theorem nfixpxy 6618
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 6600 . 2  |-  X_ x  e.  A  B  =  { z  |  ( z  Fn  { x  |  x  e.  A }  /\  A. x  e.  A  ( z `  x )  e.  B
) }
2 nfcv 2282 . . . . 5  |-  F/_ y
z
3 nftru 1443 . . . . . . 7  |-  F/ x T.
4 nfcvd 2283 . . . . . . . 8  |-  ( T. 
->  F/_ y x )
5 nfixp.1 . . . . . . . . 9  |-  F/_ y A
65a1i 9 . . . . . . . 8  |-  ( T. 
->  F/_ y A )
74, 6nfeld 2298 . . . . . . 7  |-  ( T. 
->  F/ y  x  e.  A )
83, 7nfabd 2301 . . . . . 6  |-  ( T. 
->  F/_ y { x  |  x  e.  A } )
98mptru 1341 . . . . 5  |-  F/_ y { x  |  x  e.  A }
102, 9nffn 5226 . . . 4  |-  F/ y  z  Fn  { x  |  x  e.  A }
11 df-ral 2422 . . . . 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 5440 . . . . . . . . 9  |-  ( T. 
->  F/_ y ( z `
 x ) )
14 nfixp.2 . . . . . . . . . 10  |-  F/_ y B
1514a1i 9 . . . . . . . . 9  |-  ( T. 
->  F/_ y B )
1613, 15nfeld 2298 . . . . . . . 8  |-  ( T. 
->  F/ y ( z `
 x )  e.  B )
177, 16nfimd 1565 . . . . . . 7  |-  ( T. 
->  F/ y ( x  e.  A  ->  (
z `  x )  e.  B ) )
183, 17nfald 1734 . . . . . 6  |-  ( T. 
->  F/ y A. x
( x  e.  A  ->  ( z `  x
)  e.  B ) )
1918mptru 1341 . . . . 5  |-  F/ y A. x ( x  e.  A  ->  (
z `  x )  e.  B )
2011, 19nfxfr 1451 . . . 4  |-  F/ y A. x  e.  A  ( z `  x
)  e.  B
2110, 20nfan 1545 . . 3  |-  F/ y ( z  Fn  {
x  |  x  e.  A }  /\  A. x  e.  A  (
z `  x )  e.  B )
2221nfab 2287 . 2  |-  F/_ y { z  |  ( z  Fn  { x  |  x  e.  A }  /\  A. x  e.  A  ( z `  x )  e.  B
) }
231, 22nfcxfr 2279 1  |-  F/_ y X_ x  e.  A  B
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   A.wal 1330   T. wtru 1333   F/wnf 1437    e. wcel 1481   {cab 2126   F/_wnfc 2269   A.wral 2417    Fn wfn 5125   ` cfv 5130   X_cixp 6599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3079  df-in 3081  df-ss 3088  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-opab 3997  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-iota 5095  df-fun 5132  df-fn 5133  df-fv 5138  df-ixp 6600
This theorem is referenced by: (None)
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