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Theorem disjdsct 27344
Description: A disjoint collection is distinct, i.e. each set in this collection is different of all others, provided that it does not contain the empty set This can be expressed as "the converse of the mapping function is a function", or "the mapping function is single-rooted". (Cf. funcnv 5654) (Contributed by Thierry Arnoux, 28-Feb-2017.)
Hypotheses
Ref Expression
disjdsct.0  |-  F/ x ph
disjdsct.1  |-  F/_ x A
disjdsct.2  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  ( V  \  { (/)
} ) )
disjdsct.3  |-  ( ph  -> Disj  x  e.  A  B
)
Assertion
Ref Expression
disjdsct  |-  ( ph  ->  Fun  `' ( x  e.  A  |->  B ) )
Distinct variable group:    x, V
Allowed substitution hints:    ph( x)    A( x)    B( x)

Proof of Theorem disjdsct
Dummy variables  i 
j are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 disjdsct.3 . . . . . . . 8  |-  ( ph  -> Disj  x  e.  A  B
)
2 disjdsct.1 . . . . . . . . 9  |-  F/_ x A
32disjorsf 27264 . . . . . . . 8  |-  (Disj  x  e.  A  B  <->  A. i  e.  A  A. j  e.  A  ( i  =  j  \/  ( [_ i  /  x ]_ B  i^i  [_ j  /  x ]_ B )  =  (/) ) )
41, 3sylib 196 . . . . . . 7  |-  ( ph  ->  A. i  e.  A  A. j  e.  A  ( i  =  j  \/  ( [_ i  /  x ]_ B  i^i  [_ j  /  x ]_ B )  =  (/) ) )
54r19.21bi 2836 . . . . . 6  |-  ( (
ph  /\  i  e.  A )  ->  A. j  e.  A  ( i  =  j  \/  ( [_ i  /  x ]_ B  i^i  [_ j  /  x ]_ B )  =  (/) ) )
65r19.21bi 2836 . . . . 5  |-  ( ( ( ph  /\  i  e.  A )  /\  j  e.  A )  ->  (
i  =  j  \/  ( [_ i  /  x ]_ B  i^i  [_ j  /  x ]_ B )  =  (/) ) )
7 simpr3 1004 . . . . . . . . 9  |-  ( (
ph  /\  ( i  e.  A  /\  j  e.  A  /\  ( [_ i  /  x ]_ B  i^i  [_ j  /  x ]_ B )  =  (/) ) )  -> 
( [_ i  /  x ]_ B  i^i  [_ j  /  x ]_ B )  =  (/) )
8 disjdsct.2 . . . . . . . . . . . . 13  |-  ( (
ph  /\  x  e.  A )  ->  B  e.  ( V  \  { (/)
} ) )
9 eldifsni 4159 . . . . . . . . . . . . 13  |-  ( B  e.  ( V  \  { (/) } )  ->  B  =/=  (/) )
108, 9syl 16 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  A )  ->  B  =/=  (/) )
1110sbimi 1717 . . . . . . . . . . 11  |-  ( [ i  /  x ]
( ph  /\  x  e.  A )  ->  [ i  /  x ] B  =/=  (/) )
12 sban 2114 . . . . . . . . . . . 12  |-  ( [ i  /  x ]
( ph  /\  x  e.  A )  <->  ( [
i  /  x ] ph  /\  [ i  /  x ] x  e.  A
) )
13 disjdsct.0 . . . . . . . . . . . . . 14  |-  F/ x ph
1413sbf 2094 . . . . . . . . . . . . 13  |-  ( [ i  /  x ] ph 
<-> 
ph )
152clelsb3f 27202 . . . . . . . . . . . . 13  |-  ( [ i  /  x ]
x  e.  A  <->  i  e.  A )
1614, 15anbi12i 697 . . . . . . . . . . . 12  |-  ( ( [ i  /  x ] ph  /\  [ i  /  x ] x  e.  A )  <->  ( ph  /\  i  e.  A ) )
1712, 16bitri 249 . . . . . . . . . . 11  |-  ( [ i  /  x ]
( ph  /\  x  e.  A )  <->  ( ph  /\  i  e.  A ) )
18 sbsbc 3340 . . . . . . . . . . . 12  |-  ( [ i  /  x ] B  =/=  (/)  <->  [. i  /  x ]. B  =/=  (/) )
19 sbcne12 3832 . . . . . . . . . . . 12  |-  ( [. i  /  x ]. B  =/=  (/)  <->  [_ i  /  x ]_ B  =/=  [_ i  /  x ]_ (/) )
20 csb0 3827 . . . . . . . . . . . . 13  |-  [_ i  /  x ]_ (/)  =  (/)
2120neeq2i 2754 . . . . . . . . . . . 12  |-  ( [_ i  /  x ]_ B  =/=  [_ i  /  x ]_ (/)  <->  [_ i  /  x ]_ B  =/=  (/) )
2218, 19, 213bitri 271 . . . . . . . . . . 11  |-  ( [ i  /  x ] B  =/=  (/)  <->  [_ i  /  x ]_ B  =/=  (/) )
2311, 17, 223imtr3i 265 . . . . . . . . . 10  |-  ( (
ph  /\  i  e.  A )  ->  [_ i  /  x ]_ B  =/=  (/) )
24233ad2antr1 1161 . . . . . . . . 9  |-  ( (
ph  /\  ( i  e.  A  /\  j  e.  A  /\  ( [_ i  /  x ]_ B  i^i  [_ j  /  x ]_ B )  =  (/) ) )  ->  [_ i  /  x ]_ B  =/=  (/) )
25 disj3 3876 . . . . . . . . . . . . 13  |-  ( (
[_ i  /  x ]_ B  i^i  [_ j  /  x ]_ B )  =  (/)  <->  [_ i  /  x ]_ B  =  ( [_ i  /  x ]_ B  \  [_ j  /  x ]_ B ) )
2625biimpi 194 . . . . . . . . . . . 12  |-  ( (
[_ i  /  x ]_ B  i^i  [_ j  /  x ]_ B )  =  (/)  ->  [_ i  /  x ]_ B  =  ( [_ i  /  x ]_ B  \  [_ j  /  x ]_ B ) )
2726neeq1d 2744 . . . . . . . . . . 11  |-  ( (
[_ i  /  x ]_ B  i^i  [_ j  /  x ]_ B )  =  (/)  ->  ( [_ i  /  x ]_ B  =/=  (/)  <->  ( [_ i  /  x ]_ B  \  [_ j  /  x ]_ B )  =/=  (/) ) )
2827biimpa 484 . . . . . . . . . 10  |-  ( ( ( [_ i  /  x ]_ B  i^i  [_ j  /  x ]_ B )  =  (/)  /\  [_ i  /  x ]_ B  =/=  (/) )  ->  ( [_ i  /  x ]_ B  \ 
[_ j  /  x ]_ B )  =/=  (/) )
29 difneqnul 27238 . . . . . . . . . 10  |-  ( (
[_ i  /  x ]_ B  \  [_ j  /  x ]_ B )  =/=  (/)  ->  [_ i  /  x ]_ B  =/=  [_ j  /  x ]_ B
)
3028, 29syl 16 . . . . . . . . 9  |-  ( ( ( [_ i  /  x ]_ B  i^i  [_ j  /  x ]_ B )  =  (/)  /\  [_ i  /  x ]_ B  =/=  (/) )  ->  [_ i  /  x ]_ B  =/=  [_ j  /  x ]_ B )
317, 24, 30syl2anc 661 . . . . . . . 8  |-  ( (
ph  /\  ( i  e.  A  /\  j  e.  A  /\  ( [_ i  /  x ]_ B  i^i  [_ j  /  x ]_ B )  =  (/) ) )  ->  [_ i  /  x ]_ B  =/=  [_ j  /  x ]_ B )
32313anassrs 1218 . . . . . . 7  |-  ( ( ( ( ph  /\  i  e.  A )  /\  j  e.  A
)  /\  ( [_ i  /  x ]_ B  i^i  [_ j  /  x ]_ B )  =  (/) )  ->  [_ i  /  x ]_ B  =/=  [_ j  /  x ]_ B )
3332ex 434 . . . . . 6  |-  ( ( ( ph  /\  i  e.  A )  /\  j  e.  A )  ->  (
( [_ i  /  x ]_ B  i^i  [_ j  /  x ]_ B )  =  (/)  ->  [_ i  /  x ]_ B  =/=  [_ j  /  x ]_ B ) )
3433orim2d 838 . . . . 5  |-  ( ( ( ph  /\  i  e.  A )  /\  j  e.  A )  ->  (
( i  =  j  \/  ( [_ i  /  x ]_ B  i^i  [_ j  /  x ]_ B )  =  (/) )  ->  ( i  =  j  \/  [_ i  /  x ]_ B  =/=  [_ j  /  x ]_ B ) ) )
356, 34mpd 15 . . . 4  |-  ( ( ( ph  /\  i  e.  A )  /\  j  e.  A )  ->  (
i  =  j  \/ 
[_ i  /  x ]_ B  =/=  [_ j  /  x ]_ B ) )
3635ralrimiva 2881 . . 3  |-  ( (
ph  /\  i  e.  A )  ->  A. j  e.  A  ( i  =  j  \/  [_ i  /  x ]_ B  =/=  [_ j  /  x ]_ B ) )
3736ralrimiva 2881 . 2  |-  ( ph  ->  A. i  e.  A  A. j  e.  A  ( i  =  j  \/  [_ i  /  x ]_ B  =/=  [_ j  /  x ]_ B
) )
38 nfmpt1 4542 . . 3  |-  F/_ x
( x  e.  A  |->  B )
39 eqid 2467 . . 3  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
4013, 2, 38, 39, 8funcnv4mpt 27335 . 2  |-  ( ph  ->  ( Fun  `' ( x  e.  A  |->  B )  <->  A. i  e.  A  A. j  e.  A  ( i  =  j  \/  [_ i  /  x ]_ B  =/=  [_ j  /  x ]_ B
) ) )
4137, 40mpbird 232 1  |-  ( ph  ->  Fun  `' ( x  e.  A  |->  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1379   F/wnf 1599   [wsb 1711    e. wcel 1767   F/_wnfc 2615    =/= wne 2662   A.wral 2817   [.wsbc 3336   [_csb 3440    \ cdif 3478    i^i cin 3480   (/)c0 3790   {csn 4033  Disj wdisj 4423    |-> cmpt 4511   `'ccnv 5004   Fun wfun 5588
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-fal 1385  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rmo 2825  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-disj 4424  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-fv 5602
This theorem is referenced by:  measvunilem  28008
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