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Theorem invdisj 3994
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 2519 . . 3  |-  F/ y A. x  e.  A  A. y  e.  B  C  =  x
2 df-ral 2460 . . . . 5  |-  ( A. x  e.  A  A. y  e.  B  C  =  x  <->  A. x ( x  e.  A  ->  A. y  e.  B  C  =  x ) )
3 rsp 2524 . . . . . . . . 9  |-  ( A. y  e.  B  C  =  x  ->  ( y  e.  B  ->  C  =  x ) )
4 eqcom 2179 . . . . . . . . 9  |-  ( C  =  x  <->  x  =  C )
53, 4syl6ib 161 . . . . . . . 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 254 . . . . . 6  |-  ( ( x  e.  A  ->  A. y  e.  B  C  =  x )  ->  ( ( x  e.  A  /\  y  e.  B )  ->  x  =  C ) )
87alimi 1455 . . . . 5  |-  ( A. x ( x  e.  A  ->  A. y  e.  B  C  =  x )  ->  A. x
( ( x  e.  A  /\  y  e.  B )  ->  x  =  C ) )
92, 8sylbi 121 . . . 4  |-  ( A. x  e.  A  A. y  e.  B  C  =  x  ->  A. x
( ( x  e.  A  /\  y  e.  B )  ->  x  =  C ) )
10 mo2icl 2916 . . . 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 1522 . 2  |-  ( A. x  e.  A  A. y  e.  B  C  =  x  ->  A. y E* x ( x  e.  A  /\  y  e.  B ) )
13 dfdisj2 3979 . 2  |-  (Disj  x  e.  A  B  <->  A. y E* x ( x  e.  A  /\  y  e.  B ) )
1412, 13sylibr 134 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 104   A.wal 1351    = wceq 1353   E*wmo 2027    e. wcel 2148   A.wral 2455  Disj wdisj 3977
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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rmo 2463  df-v 2739  df-disj 3978
This theorem is referenced by:  phisum  12220
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