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Theorem invdisj 3831
Description: If there is a function  C (
y ) such that  C (
y )  =  x for all  y  e.  B
( x ), then the sets  B ( x ) for distinct  x  e.  A are disjoint. (Contributed by Mario Carneiro, 10-Dec-2016.)
Assertion
Ref Expression
invdisj  |-  ( A. x  e.  A  A. y  e.  B  C  =  x  -> Disj  x  e.  A  B )
Distinct variable groups:    x, y    y, A    y, B    x, C
Allowed substitution hints:    A( x)    B( x)    C( y)

Proof of Theorem invdisj
StepHypRef Expression
1 nfra2xy 2418 . . 3  |-  F/ y A. x  e.  A  A. y  e.  B  C  =  x
2 df-ral 2364 . . . . 5  |-  ( A. x  e.  A  A. y  e.  B  C  =  x  <->  A. x ( x  e.  A  ->  A. y  e.  B  C  =  x ) )
3 rsp 2423 . . . . . . . . 9  |-  ( A. y  e.  B  C  =  x  ->  ( y  e.  B  ->  C  =  x ) )
4 eqcom 2090 . . . . . . . . 9  |-  ( C  =  x  <->  x  =  C )
53, 4syl6ib 159 . . . . . . . 8  |-  ( A. y  e.  B  C  =  x  ->  ( y  e.  B  ->  x  =  C ) )
65imim2i 12 . . . . . . 7  |-  ( ( x  e.  A  ->  A. y  e.  B  C  =  x )  ->  ( x  e.  A  ->  ( y  e.  B  ->  x  =  C ) ) )
76impd 251 . . . . . 6  |-  ( ( x  e.  A  ->  A. y  e.  B  C  =  x )  ->  ( ( x  e.  A  /\  y  e.  B )  ->  x  =  C ) )
87alimi 1389 . . . . 5  |-  ( A. x ( x  e.  A  ->  A. y  e.  B  C  =  x )  ->  A. x
( ( x  e.  A  /\  y  e.  B )  ->  x  =  C ) )
92, 8sylbi 119 . . . 4  |-  ( A. x  e.  A  A. y  e.  B  C  =  x  ->  A. x
( ( x  e.  A  /\  y  e.  B )  ->  x  =  C ) )
10 mo2icl 2792 . . . 4  |-  ( A. x ( ( x  e.  A  /\  y  e.  B )  ->  x  =  C )  ->  E* x ( x  e.  A  /\  y  e.  B ) )
119, 10syl 14 . . 3  |-  ( A. x  e.  A  A. y  e.  B  C  =  x  ->  E* x
( x  e.  A  /\  y  e.  B
) )
121, 11alrimi 1460 . 2  |-  ( A. x  e.  A  A. y  e.  B  C  =  x  ->  A. y E* x ( x  e.  A  /\  y  e.  B ) )
13 dfdisj2 3816 . 2  |-  (Disj  x  e.  A  B  <->  A. y E* x ( x  e.  A  /\  y  e.  B ) )
1412, 13sylibr 132 1  |-  ( A. x  e.  A  A. y  e.  B  C  =  x  -> Disj  x  e.  A  B )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102   A.wal 1287    = wceq 1289    e. wcel 1438   E*wmo 1949   A.wral 2359  Disj wdisj 3814
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-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  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-rmo 2367  df-v 2621  df-disj 3815
This theorem is referenced by: (None)
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