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Theorem invdisj 4193
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 nfra2 2752 . . 3  |-  F/ y A. x  e.  A  A. y  e.  B  C  =  x
2 df-ral 2702 . . . . 5  |-  ( A. x  e.  A  A. y  e.  B  C  =  x  <->  A. x ( x  e.  A  ->  A. y  e.  B  C  =  x ) )
3 rsp 2758 . . . . . . . . 9  |-  ( A. y  e.  B  C  =  x  ->  ( y  e.  B  ->  C  =  x ) )
4 eqcom 2437 . . . . . . . . 9  |-  ( C  =  x  <->  x  =  C )
53, 4syl6ib 218 . . . . . . . 8  |-  ( A. y  e.  B  C  =  x  ->  ( y  e.  B  ->  x  =  C ) )
65imim2i 14 . . . . . . 7  |-  ( ( x  e.  A  ->  A. y  e.  B  C  =  x )  ->  ( x  e.  A  ->  ( y  e.  B  ->  x  =  C ) ) )
76imp3a 421 . . . . . 6  |-  ( ( x  e.  A  ->  A. y  e.  B  C  =  x )  ->  ( ( x  e.  A  /\  y  e.  B )  ->  x  =  C ) )
87alimi 1568 . . . . 5  |-  ( A. x ( x  e.  A  ->  A. y  e.  B  C  =  x )  ->  A. x
( ( x  e.  A  /\  y  e.  B )  ->  x  =  C ) )
92, 8sylbi 188 . . . 4  |-  ( A. x  e.  A  A. y  e.  B  C  =  x  ->  A. x
( ( x  e.  A  /\  y  e.  B )  ->  x  =  C ) )
10 mo2icl 3105 . . . 4  |-  ( A. x ( ( x  e.  A  /\  y  e.  B )  ->  x  =  C )  ->  E* x ( x  e.  A  /\  y  e.  B ) )
119, 10syl 16 . . 3  |-  ( A. x  e.  A  A. y  e.  B  C  =  x  ->  E* x
( x  e.  A  /\  y  e.  B
) )
121, 11alrimi 1781 . 2  |-  ( A. x  e.  A  A. y  e.  B  C  =  x  ->  A. y E* x ( x  e.  A  /\  y  e.  B ) )
13 dfdisj2 4176 . 2  |-  (Disj  x  e.  A B  <->  A. y E* x ( x  e.  A  /\  y  e.  B ) )
1412, 13sylibr 204 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 359   A.wal 1549    = wceq 1652    e. wcel 1725   E*wmo 2281   A.wral 2697  Disj wdisj 4174
This theorem is referenced by:  ackbijnn  12599  incexc2  12610  itg1addlem1  19576  musum  20968  lgsquadlem1  21130  lgsquadlem2  21131  disjabrex  24016  disjabrexf  24017  phisum  27486
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rmo 2705  df-v 2950  df-disj 4175
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