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Theorem disjdsct 25917
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 5475) (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 25843 . . . . . . . 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 2812 . . . . . 6  |-  ( (
ph  /\  i  e.  A )  ->  A. j  e.  A  ( i  =  j  \/  ( [_ i  /  x ]_ B  i^i  [_ j  /  x ]_ B )  =  (/) ) )
65r19.21bi 2812 . . . . 5  |-  ( ( ( ph  /\  i  e.  A )  /\  j  e.  A )  ->  (
i  =  j  \/  ( [_ i  /  x ]_ B  i^i  [_ j  /  x ]_ B )  =  (/) ) )
7 simpr3 991 . . . . . . . . 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 3998 . . . . . . . . . . . . 13  |-  ( B  e.  ( V  \  { (/) } )  ->  B  =/=  (/) )
108, 9syl 16 . . . . . . . . . . . 12  |-  ( (
ph  /\  x  e.  A )  ->  B  =/=  (/) )
1110sbimi 1711 . . . . . . . . . . 11  |-  ( [ i  /  x ]
( ph  /\  x  e.  A )  ->  [ i  /  x ] B  =/=  (/) )
12 sban 2097 . . . . . . . . . . . 12  |-  ( [ i  /  x ]
( ph  /\  x  e.  A )  <->  ( [
i  /  x ] ph  /\  [ i  /  x ] x  e.  A
) )
13 disjdsct.0 . . . . . . . . . . . . . 14  |-  F/ x ph
1413sbf 2076 . . . . . . . . . . . . 13  |-  ( [ i  /  x ] ph 
<-> 
ph )
152clelsb3f 25783 . . . . . . . . . . . . 13  |-  ( [ i  /  x ]
x  e.  A  <->  i  e.  A )
1614, 15anbi12i 692 . . . . . . . . . . . 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 3187 . . . . . . . . . . . 12  |-  ( [ i  /  x ] B  =/=  (/)  <->  [. i  /  x ]. B  =/=  (/) )
19 sbcne12 3676 . . . . . . . . . . . 12  |-  ( [. i  /  x ]. B  =/=  (/)  <->  [_ i  /  x ]_ B  =/=  [_ i  /  x ]_ (/) )
20 csb0 3671 . . . . . . . . . . . . 13  |-  [_ i  /  x ]_ (/)  =  (/)
2120neeq2i 2617 . . . . . . . . . . . 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 1148 . . . . . . . . 9  |-  ( (
ph  /\  ( i  e.  A  /\  j  e.  A  /\  ( [_ i  /  x ]_ B  i^i  [_ j  /  x ]_ B )  =  (/) ) )  ->  [_ i  /  x ]_ B  =/=  (/) )
25 disj3 3720 . . . . . . . . . . . . 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 2619 . . . . . . . . . . 11  |-  ( (
[_ i  /  x ]_ B  i^i  [_ j  /  x ]_ B )  =  (/)  ->  ( [_ i  /  x ]_ B  =/=  (/)  <->  ( [_ i  /  x ]_ B  \  [_ j  /  x ]_ B )  =/=  (/) ) )
2827biimpa 481 . . . . . . . . . 10  |-  ( ( ( [_ i  /  x ]_ B  i^i  [_ j  /  x ]_ B )  =  (/)  /\  [_ i  /  x ]_ B  =/=  (/) )  ->  ( [_ i  /  x ]_ B  \ 
[_ j  /  x ]_ B )  =/=  (/) )
29 difneqnul 25817 . . . . . . . . . 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 656 . . . . . . . 8  |-  ( (
ph  /\  ( i  e.  A  /\  j  e.  A  /\  ( [_ i  /  x ]_ B  i^i  [_ j  /  x ]_ B )  =  (/) ) )  ->  [_ i  /  x ]_ B  =/=  [_ j  /  x ]_ B )
32313anassrs 1204 . . . . . . 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 831 . . . . 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 2797 . . 3  |-  ( (
ph  /\  i  e.  A )  ->  A. j  e.  A  ( i  =  j  \/  [_ i  /  x ]_ B  =/=  [_ j  /  x ]_ B ) )
3736ralrimiva 2797 . 2  |-  ( ph  ->  A. i  e.  A  A. j  e.  A  ( i  =  j  \/  [_ i  /  x ]_ B  =/=  [_ j  /  x ]_ B
) )
38 nfmpt1 4378 . . 3  |-  F/_ x
( x  e.  A  |->  B )
39 eqid 2441 . . 3  |-  ( x  e.  A  |->  B )  =  ( x  e.  A  |->  B )
4013, 2, 38, 39, 8funcnv4mpt 25908 . 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 960    = wceq 1364   F/wnf 1594   [wsb 1705    e. wcel 1761   F/_wnfc 2564    =/= wne 2604   A.wral 2713   [.wsbc 3183   [_csb 3285    \ cdif 3322    i^i cin 3324   (/)c0 3634   {csn 3874  Disj wdisj 4259    e. cmpt 4347   `'ccnv 4835   Fun wfun 5409
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-sep 4410  ax-nul 4418  ax-pr 4528
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-fal 1370  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rmo 2721  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-disj 4260  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-fv 5423
This theorem is referenced by:  measvunilem  26546
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