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Theorem nfixpxy 6541
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 6523 . 2  |-  X_ x  e.  A  B  =  { z  |  ( z  Fn  { x  |  x  e.  A }  /\  A. x  e.  A  ( z `  x )  e.  B
) }
2 nfcv 2240 . . . . 5  |-  F/_ y
z
3 nftru 1410 . . . . . . 7  |-  F/ x T.
4 nfcvd 2241 . . . . . . . 8  |-  ( T. 
->  F/_ y x )
5 nfixp.1 . . . . . . . . 9  |-  F/_ y A
65a1i 9 . . . . . . . 8  |-  ( T. 
->  F/_ y A )
74, 6nfeld 2256 . . . . . . 7  |-  ( T. 
->  F/ y  x  e.  A )
83, 7nfabd 2259 . . . . . 6  |-  ( T. 
->  F/_ y { x  |  x  e.  A } )
98mptru 1308 . . . . 5  |-  F/_ y { x  |  x  e.  A }
102, 9nffn 5155 . . . 4  |-  F/ y  z  Fn  { x  |  x  e.  A }
11 df-ral 2380 . . . . 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 5365 . . . . . . . . 9  |-  ( T. 
->  F/_ y ( z `
 x ) )
14 nfixp.2 . . . . . . . . . 10  |-  F/_ y B
1514a1i 9 . . . . . . . . 9  |-  ( T. 
->  F/_ y B )
1613, 15nfeld 2256 . . . . . . . 8  |-  ( T. 
->  F/ y ( z `
 x )  e.  B )
177, 16nfimd 1532 . . . . . . 7  |-  ( T. 
->  F/ y ( x  e.  A  ->  (
z `  x )  e.  B ) )
183, 17nfald 1701 . . . . . 6  |-  ( T. 
->  F/ y A. x
( x  e.  A  ->  ( z `  x
)  e.  B ) )
1918mptru 1308 . . . . 5  |-  F/ y A. x ( x  e.  A  ->  (
z `  x )  e.  B )
2011, 19nfxfr 1418 . . . 4  |-  F/ y A. x  e.  A  ( z `  x
)  e.  B
2110, 20nfan 1512 . . 3  |-  F/ y ( z  Fn  {
x  |  x  e.  A }  /\  A. x  e.  A  (
z `  x )  e.  B )
2221nfab 2245 . 2  |-  F/_ y { z  |  ( z  Fn  { x  |  x  e.  A }  /\  A. x  e.  A  ( z `  x )  e.  B
) }
231, 22nfcxfr 2237 1  |-  F/_ y X_ x  e.  A  B
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103   A.wal 1297   T. wtru 1300   F/wnf 1404    e. wcel 1448   {cab 2086   F/_wnfc 2227   A.wral 2375    Fn wfn 5054   ` cfv 5059   X_cixp 6522
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ral 2380  df-rex 2381  df-v 2643  df-un 3025  df-in 3027  df-ss 3034  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-iota 5024  df-fun 5061  df-fn 5062  df-fv 5067  df-ixp 6523
This theorem is referenced by: (None)
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