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Theorem disjxp1 6001
Description: The sets of a cartesian product are disjoint if the sets in the first argument are disjoint. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
Hypothesis
Ref Expression
disjxp1.1  |-  ( ph  -> Disj  x  e.  A  B
)
Assertion
Ref Expression
disjxp1  |-  ( ph  -> Disj  x  e.  A  ( B  X.  C ) )
Distinct variable groups:    x, A    ph, x
Allowed substitution hints:    B( x)    C( x)

Proof of Theorem disjxp1
Dummy variables  y  z  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 xp1st 5936 . . . . . . 7  |-  ( y  e.  ( B  X.  C )  ->  ( 1st `  y )  e.  B )
2 xp1st 5936 . . . . . . 7  |-  ( y  e.  ( [_ w  /  x ]_ B  X.  [_ w  /  x ]_ C )  ->  ( 1st `  y )  e. 
[_ w  /  x ]_ B )
3 disjxp1.1 . . . . . . . . . . . 12  |-  ( ph  -> Disj  x  e.  A  B
)
4 df-disj 3823 . . . . . . . . . . . 12  |-  (Disj  x  e.  A  B  <->  A. z E* x  e.  A  z  e.  B )
53, 4sylib 120 . . . . . . . . . . 11  |-  ( ph  ->  A. z E* x  e.  A  z  e.  B )
6 1stexg 5938 . . . . . . . . . . . . 13  |-  ( y  e.  _V  ->  ( 1st `  y )  e. 
_V )
76elv 2623 . . . . . . . . . . . 12  |-  ( 1st `  y )  e.  _V
8 eleq1 2150 . . . . . . . . . . . . 13  |-  ( z  =  ( 1st `  y
)  ->  ( z  e.  B  <->  ( 1st `  y
)  e.  B ) )
98rmobidv 2555 . . . . . . . . . . . 12  |-  ( z  =  ( 1st `  y
)  ->  ( E* x  e.  A  z  e.  B  <->  E* x  e.  A  ( 1st `  y )  e.  B ) )
107, 9spcv 2712 . . . . . . . . . . 11  |-  ( A. z E* x  e.  A  z  e.  B  ->  E* x  e.  A  ( 1st `  y )  e.  B )
115, 10syl 14 . . . . . . . . . 10  |-  ( ph  ->  E* x  e.  A  ( 1st `  y )  e.  B )
12 nfcv 2228 . . . . . . . . . . 11  |-  F/_ x A
13 nfcv 2228 . . . . . . . . . . 11  |-  F/_ w A
14 nfcsb1v 2963 . . . . . . . . . . . 12  |-  F/_ x [_ w  /  x ]_ B
1514nfel2 2241 . . . . . . . . . . 11  |-  F/ x
( 1st `  y
)  e.  [_ w  /  x ]_ B
16 csbeq1a 2941 . . . . . . . . . . . 12  |-  ( x  =  w  ->  B  =  [_ w  /  x ]_ B )
1716eleq2d 2157 . . . . . . . . . . 11  |-  ( x  =  w  ->  (
( 1st `  y
)  e.  B  <->  ( 1st `  y )  e.  [_ w  /  x ]_ B
) )
1812, 13, 15, 17rmo4f 2813 . . . . . . . . . 10  |-  ( E* x  e.  A  ( 1st `  y )  e.  B  <->  A. x  e.  A  A. w  e.  A  ( (
( 1st `  y
)  e.  B  /\  ( 1st `  y )  e.  [_ w  /  x ]_ B )  ->  x  =  w )
)
1911, 18sylib 120 . . . . . . . . 9  |-  ( ph  ->  A. x  e.  A  A. w  e.  A  ( ( ( 1st `  y )  e.  B  /\  ( 1st `  y
)  e.  [_ w  /  x ]_ B )  ->  x  =  w ) )
2019r19.21bi 2461 . . . . . . . 8  |-  ( (
ph  /\  x  e.  A )  ->  A. w  e.  A  ( (
( 1st `  y
)  e.  B  /\  ( 1st `  y )  e.  [_ w  /  x ]_ B )  ->  x  =  w )
)
2120r19.21bi 2461 . . . . . . 7  |-  ( ( ( ph  /\  x  e.  A )  /\  w  e.  A )  ->  (
( ( 1st `  y
)  e.  B  /\  ( 1st `  y )  e.  [_ w  /  x ]_ B )  ->  x  =  w )
)
221, 2, 21syl2ani 400 . . . . . 6  |-  ( ( ( ph  /\  x  e.  A )  /\  w  e.  A )  ->  (
( y  e.  ( B  X.  C )  /\  y  e.  (
[_ w  /  x ]_ B  X.  [_ w  /  x ]_ C ) )  ->  x  =  w ) )
2322ralrimiva 2446 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  A. w  e.  A  ( (
y  e.  ( B  X.  C )  /\  y  e.  ( [_ w  /  x ]_ B  X.  [_ w  /  x ]_ C ) )  ->  x  =  w )
)
2423ralrimiva 2446 . . . 4  |-  ( ph  ->  A. x  e.  A  A. w  e.  A  ( ( y  e.  ( B  X.  C
)  /\  y  e.  ( [_ w  /  x ]_ B  X.  [_ w  /  x ]_ C ) )  ->  x  =  w ) )
25 nfcsb1v 2963 . . . . . . 7  |-  F/_ x [_ w  /  x ]_ C
2614, 25nfxp 4464 . . . . . 6  |-  F/_ x
( [_ w  /  x ]_ B  X.  [_ w  /  x ]_ C )
2726nfel2 2241 . . . . 5  |-  F/ x  y  e.  ( [_ w  /  x ]_ B  X.  [_ w  /  x ]_ C )
28 csbeq1a 2941 . . . . . . 7  |-  ( x  =  w  ->  C  =  [_ w  /  x ]_ C )
2916, 28xpeq12d 4463 . . . . . 6  |-  ( x  =  w  ->  ( B  X.  C )  =  ( [_ w  /  x ]_ B  X.  [_ w  /  x ]_ C
) )
3029eleq2d 2157 . . . . 5  |-  ( x  =  w  ->  (
y  e.  ( B  X.  C )  <->  y  e.  ( [_ w  /  x ]_ B  X.  [_ w  /  x ]_ C ) ) )
3112, 13, 27, 30rmo4f 2813 . . . 4  |-  ( E* x  e.  A  y  e.  ( B  X.  C )  <->  A. x  e.  A  A. w  e.  A  ( (
y  e.  ( B  X.  C )  /\  y  e.  ( [_ w  /  x ]_ B  X.  [_ w  /  x ]_ C ) )  ->  x  =  w )
)
3224, 31sylibr 132 . . 3  |-  ( ph  ->  E* x  e.  A  y  e.  ( B  X.  C ) )
3332alrimiv 1802 . 2  |-  ( ph  ->  A. y E* x  e.  A  y  e.  ( B  X.  C
) )
34 df-disj 3823 . 2  |-  (Disj  x  e.  A  ( B  X.  C )  <->  A. y E* x  e.  A  y  e.  ( B  X.  C ) )
3533, 34sylibr 132 1  |-  ( ph  -> Disj  x  e.  A  ( B  X.  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102   A.wal 1287    = wceq 1289    e. wcel 1438   A.wral 2359   E*wrmo 2362   _Vcvv 2619   [_csb 2933  Disj wdisj 3822    X. cxp 4436   ` cfv 5015   1stc1st 5909
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rmo 2367  df-v 2621  df-sbc 2841  df-csb 2934  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-disj 3823  df-br 3846  df-opab 3900  df-mpt 3901  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-iota 4980  df-fun 5017  df-fn 5018  df-f 5019  df-fo 5021  df-fv 5023  df-1st 5911
This theorem is referenced by:  disjsnxp  6002
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