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Theorem invdisj 4356
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 2769 . . 3  |-  F/ y A. x  e.  A  A. y  e.  B  C  =  x
2 df-ral 2737 . . . . 5  |-  ( A. x  e.  A  A. y  e.  B  C  =  x  <->  A. x ( x  e.  A  ->  A. y  e.  B  C  =  x ) )
3 rsp 2748 . . . . . . . . 9  |-  ( A. y  e.  B  C  =  x  ->  ( y  e.  B  ->  C  =  x ) )
4 eqcom 2391 . . . . . . . . 9  |-  ( C  =  x  <->  x  =  C )
53, 4syl6ib 226 . . . . . . . 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 ) ) )
76impd 429 . . . . . 6  |-  ( ( x  e.  A  ->  A. y  e.  B  C  =  x )  ->  ( ( x  e.  A  /\  y  e.  B )  ->  x  =  C ) )
87alimi 1641 . . . . 5  |-  ( A. x ( x  e.  A  ->  A. y  e.  B  C  =  x )  ->  A. x
( ( x  e.  A  /\  y  e.  B )  ->  x  =  C ) )
92, 8sylbi 195 . . . 4  |-  ( A. x  e.  A  A. y  e.  B  C  =  x  ->  A. x
( ( x  e.  A  /\  y  e.  B )  ->  x  =  C ) )
10 mo2icl 3203 . . . 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 1885 . 2  |-  ( A. x  e.  A  A. y  e.  B  C  =  x  ->  A. y E* x ( x  e.  A  /\  y  e.  B ) )
13 dfdisj2 4340 . 2  |-  (Disj  x  e.  A  B  <->  A. y E* x ( x  e.  A  /\  y  e.  B ) )
1412, 13sylibr 212 1  |-  ( A. x  e.  A  A. y  e.  B  C  =  x  -> Disj  x  e.  A  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367   A.wal 1397    = wceq 1399    e. wcel 1826   E*wmo 2219   A.wral 2732  Disj wdisj 4338
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ral 2737  df-rmo 2740  df-v 3036  df-disj 4339
This theorem is referenced by:  invdisjrab  4357  ackbijnn  13642  incexc2  13652  itg1addlem1  22184  musum  23584  lgsquadlem1  23746  lgsquadlem2  23747  disjabrex  27572  disjabrexf  27573  phisum  31327
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