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Theorem imadifxp 27131
Description: Image of the difference with a Cartesian product (Contributed by Thierry Arnoux, 13-Dec-2017.)
Assertion
Ref Expression
imadifxp  |-  ( C 
C_  A  ->  (
( R  \  ( A  X.  B ) )
" C )  =  ( ( R " C )  \  B
) )

Proof of Theorem imadifxp
StepHypRef Expression
1 ima0 5350 . . . 4  |-  ( ( R  \  ( A  X.  B ) )
" (/) )  =  (/)
2 imaeq2 5331 . . . 4  |-  ( C  =  (/)  ->  ( ( R  \  ( A  X.  B ) )
" C )  =  ( ( R  \ 
( A  X.  B
) ) " (/) ) )
3 imaeq2 5331 . . . . . . 7  |-  ( C  =  (/)  ->  ( R
" C )  =  ( R " (/) ) )
4 ima0 5350 . . . . . . 7  |-  ( R
" (/) )  =  (/)
53, 4syl6eq 2524 . . . . . 6  |-  ( C  =  (/)  ->  ( R
" C )  =  (/) )
65difeq1d 3621 . . . . 5  |-  ( C  =  (/)  ->  ( ( R " C ) 
\  B )  =  ( (/)  \  B ) )
7 0dif 3898 . . . . 5  |-  ( (/)  \  B )  =  (/)
86, 7syl6eq 2524 . . . 4  |-  ( C  =  (/)  ->  ( ( R " C ) 
\  B )  =  (/) )
91, 2, 83eqtr4a 2534 . . 3  |-  ( C  =  (/)  ->  ( ( R  \  ( A  X.  B ) )
" C )  =  ( ( R " C )  \  B
) )
109adantl 466 . 2  |-  ( ( C  C_  A  /\  C  =  (/) )  -> 
( ( R  \ 
( A  X.  B
) ) " C
)  =  ( ( R " C ) 
\  B ) )
11 inundif 3905 . . . . . . . . 9  |-  ( ( R  i^i  ( A  X.  B ) )  u.  ( R  \ 
( A  X.  B
) ) )  =  R
1211imaeq1i 5332 . . . . . . . 8  |-  ( ( ( R  i^i  ( A  X.  B ) )  u.  ( R  \ 
( A  X.  B
) ) ) " C )  =  ( R " C )
13 imaundir 5417 . . . . . . . 8  |-  ( ( ( R  i^i  ( A  X.  B ) )  u.  ( R  \ 
( A  X.  B
) ) ) " C )  =  ( ( ( R  i^i  ( A  X.  B
) ) " C
)  u.  ( ( R  \  ( A  X.  B ) )
" C ) )
1412, 13eqtr3i 2498 . . . . . . 7  |-  ( R
" C )  =  ( ( ( R  i^i  ( A  X.  B ) ) " C )  u.  (
( R  \  ( A  X.  B ) )
" C ) )
1514difeq1i 3618 . . . . . 6  |-  ( ( R " C ) 
\  B )  =  ( ( ( ( R  i^i  ( A  X.  B ) )
" C )  u.  ( ( R  \ 
( A  X.  B
) ) " C
) )  \  B
)
16 difundir 3751 . . . . . 6  |-  ( ( ( ( R  i^i  ( A  X.  B
) ) " C
)  u.  ( ( R  \  ( A  X.  B ) )
" C ) ) 
\  B )  =  ( ( ( ( R  i^i  ( A  X.  B ) )
" C )  \  B )  u.  (
( ( R  \ 
( A  X.  B
) ) " C
)  \  B )
)
1715, 16eqtri 2496 . . . . 5  |-  ( ( R " C ) 
\  B )  =  ( ( ( ( R  i^i  ( A  X.  B ) )
" C )  \  B )  u.  (
( ( R  \ 
( A  X.  B
) ) " C
)  \  B )
)
18 inss2 3719 . . . . . . . . 9  |-  ( R  i^i  ( A  X.  B ) )  C_  ( A  X.  B
)
19 imass1 5369 . . . . . . . . 9  |-  ( ( R  i^i  ( A  X.  B ) ) 
C_  ( A  X.  B )  ->  (
( R  i^i  ( A  X.  B ) )
" C )  C_  ( ( A  X.  B ) " C
) )
20 ssdif 3639 . . . . . . . . 9  |-  ( ( ( R  i^i  ( A  X.  B ) )
" C )  C_  ( ( A  X.  B ) " C
)  ->  ( (
( R  i^i  ( A  X.  B ) )
" C )  \  B )  C_  (
( ( A  X.  B ) " C
)  \  B )
)
2118, 19, 20mp2b 10 . . . . . . . 8  |-  ( ( ( R  i^i  ( A  X.  B ) )
" C )  \  B )  C_  (
( ( A  X.  B ) " C
)  \  B )
22 xpima 5447 . . . . . . . . . . 11  |-  ( ( A  X.  B )
" C )  =  if ( ( A  i^i  C )  =  (/) ,  (/) ,  B )
23 incom 3691 . . . . . . . . . . . . . . 15  |-  ( C  i^i  A )  =  ( A  i^i  C
)
24 df-ss 3490 . . . . . . . . . . . . . . . 16  |-  ( C 
C_  A  <->  ( C  i^i  A )  =  C )
2524biimpi 194 . . . . . . . . . . . . . . 15  |-  ( C 
C_  A  ->  ( C  i^i  A )  =  C )
2623, 25syl5eqr 2522 . . . . . . . . . . . . . 14  |-  ( C 
C_  A  ->  ( A  i^i  C )  =  C )
2726adantl 466 . . . . . . . . . . . . 13  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  ( A  i^i  C )  =  C )
28 simpl 457 . . . . . . . . . . . . 13  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  C  =/=  (/) )
2927, 28eqnetrd 2760 . . . . . . . . . . . 12  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  ( A  i^i  C )  =/=  (/) )
30 df-ne 2664 . . . . . . . . . . . . 13  |-  ( ( A  i^i  C )  =/=  (/)  <->  -.  ( A  i^i  C )  =  (/) )
3130biimpi 194 . . . . . . . . . . . 12  |-  ( ( A  i^i  C )  =/=  (/)  ->  -.  ( A  i^i  C )  =  (/) )
32 iffalse 3948 . . . . . . . . . . . 12  |-  ( -.  ( A  i^i  C
)  =  (/)  ->  if ( ( A  i^i  C )  =  (/) ,  (/) ,  B )  =  B )
3329, 31, 323syl 20 . . . . . . . . . . 11  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  if ( ( A  i^i  C )  =  (/) ,  (/) ,  B )  =  B )
3422, 33syl5eq 2520 . . . . . . . . . 10  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  (
( A  X.  B
) " C )  =  B )
3534difeq1d 3621 . . . . . . . . 9  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  (
( ( A  X.  B ) " C
)  \  B )  =  ( B  \  B ) )
36 difid 3895 . . . . . . . . 9  |-  ( B 
\  B )  =  (/)
3735, 36syl6eq 2524 . . . . . . . 8  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  (
( ( A  X.  B ) " C
)  \  B )  =  (/) )
3821, 37syl5sseq 3552 . . . . . . 7  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  (
( ( R  i^i  ( A  X.  B
) ) " C
)  \  B )  C_  (/) )
39 ss0 3816 . . . . . . 7  |-  ( ( ( ( R  i^i  ( A  X.  B
) ) " C
)  \  B )  C_  (/)  ->  ( ( ( R  i^i  ( A  X.  B ) )
" C )  \  B )  =  (/) )
4038, 39syl 16 . . . . . 6  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  (
( ( R  i^i  ( A  X.  B
) ) " C
)  \  B )  =  (/) )
41 df-ima 5012 . . . . . . . . . . 11  |-  ( ( R  \  ( A  X.  B ) )
" C )  =  ran  ( ( R 
\  ( A  X.  B ) )  |`  C )
42 df-res 5011 . . . . . . . . . . . 12  |-  ( ( R  \  ( A  X.  B ) )  |`  C )  =  ( ( R  \  ( A  X.  B ) )  i^i  ( C  X.  _V ) )
4342rneqi 5227 . . . . . . . . . . 11  |-  ran  (
( R  \  ( A  X.  B ) )  |`  C )  =  ran  ( ( R  \ 
( A  X.  B
) )  i^i  ( C  X.  _V ) )
4441, 43eqtri 2496 . . . . . . . . . 10  |-  ( ( R  \  ( A  X.  B ) )
" C )  =  ran  ( ( R 
\  ( A  X.  B ) )  i^i  ( C  X.  _V ) )
4544ineq1i 3696 . . . . . . . . 9  |-  ( ( ( R  \  ( A  X.  B ) )
" C )  i^i 
B )  =  ( ran  ( ( R 
\  ( A  X.  B ) )  i^i  ( C  X.  _V ) )  i^i  B
)
46 xpss1 5109 . . . . . . . . . . . 12  |-  ( C 
C_  A  ->  ( C  X.  _V )  C_  ( A  X.  _V )
)
47 sslin 3724 . . . . . . . . . . . 12  |-  ( ( C  X.  _V )  C_  ( A  X.  _V )  ->  ( ( R 
\  ( A  X.  B ) )  i^i  ( C  X.  _V ) )  C_  (
( R  \  ( A  X.  B ) )  i^i  ( A  X.  _V ) ) )
48 rnss 5229 . . . . . . . . . . . 12  |-  ( ( ( R  \  ( A  X.  B ) )  i^i  ( C  X.  _V ) )  C_  (
( R  \  ( A  X.  B ) )  i^i  ( A  X.  _V ) )  ->  ran  ( ( R  \ 
( A  X.  B
) )  i^i  ( C  X.  _V ) ) 
C_  ran  ( ( R  \  ( A  X.  B ) )  i^i  ( A  X.  _V ) ) )
4946, 47, 483syl 20 . . . . . . . . . . 11  |-  ( C 
C_  A  ->  ran  ( ( R  \ 
( A  X.  B
) )  i^i  ( C  X.  _V ) ) 
C_  ran  ( ( R  \  ( A  X.  B ) )  i^i  ( A  X.  _V ) ) )
5049adantl 466 . . . . . . . . . 10  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  ran  ( ( R  \ 
( A  X.  B
) )  i^i  ( C  X.  _V ) ) 
C_  ran  ( ( R  \  ( A  X.  B ) )  i^i  ( A  X.  _V ) ) )
51 ssn0 3818 . . . . . . . . . . . 12  |-  ( ( C  C_  A  /\  C  =/=  (/) )  ->  A  =/=  (/) )
5251ancoms 453 . . . . . . . . . . 11  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  A  =/=  (/) )
53 inss1 3718 . . . . . . . . . . . . . . . 16  |-  ( ( A  X.  _V )  i^i  R )  C_  ( A  X.  _V )
54 ssdif 3639 . . . . . . . . . . . . . . . 16  |-  ( ( ( A  X.  _V )  i^i  R )  C_  ( A  X.  _V )  ->  ( ( ( A  X.  _V )  i^i 
R )  \  ( A  X.  B ) ) 
C_  ( ( A  X.  _V )  \ 
( A  X.  B
) ) )
5553, 54ax-mp 5 . . . . . . . . . . . . . . 15  |-  ( ( ( A  X.  _V )  i^i  R )  \ 
( A  X.  B
) )  C_  (
( A  X.  _V )  \  ( A  X.  B ) )
56 incom 3691 . . . . . . . . . . . . . . . 16  |-  ( ( A  X.  _V )  i^i  ( R  \  ( A  X.  B ) ) )  =  ( ( R  \  ( A  X.  B ) )  i^i  ( A  X.  _V ) )
57 indif2 3741 . . . . . . . . . . . . . . . 16  |-  ( ( A  X.  _V )  i^i  ( R  \  ( A  X.  B ) ) )  =  ( ( ( A  X.  _V )  i^i  R )  \ 
( A  X.  B
) )
5856, 57eqtr3i 2498 . . . . . . . . . . . . . . 15  |-  ( ( R  \  ( A  X.  B ) )  i^i  ( A  X.  _V ) )  =  ( ( ( A  X.  _V )  i^i  R ) 
\  ( A  X.  B ) )
59 difxp2 5431 . . . . . . . . . . . . . . 15  |-  ( A  X.  ( _V  \  B ) )  =  ( ( A  X.  _V )  \  ( A  X.  B ) )
6055, 58, 593sstr4i 3543 . . . . . . . . . . . . . 14  |-  ( ( R  \  ( A  X.  B ) )  i^i  ( A  X.  _V ) )  C_  ( A  X.  ( _V  \  B ) )
61 rnss 5229 . . . . . . . . . . . . . 14  |-  ( ( ( R  \  ( A  X.  B ) )  i^i  ( A  X.  _V ) )  C_  ( A  X.  ( _V  \  B ) )  ->  ran  ( ( R  \ 
( A  X.  B
) )  i^i  ( A  X.  _V ) ) 
C_  ran  ( A  X.  ( _V  \  B
) ) )
6260, 61mp1i 12 . . . . . . . . . . . . 13  |-  ( A  =/=  (/)  ->  ran  ( ( R  \  ( A  X.  B ) )  i^i  ( A  X.  _V ) )  C_  ran  ( A  X.  ( _V  \  B ) ) )
63 rnxp 5435 . . . . . . . . . . . . 13  |-  ( A  =/=  (/)  ->  ran  ( A  X.  ( _V  \  B ) )  =  ( _V  \  B
) )
6462, 63sseqtrd 3540 . . . . . . . . . . . 12  |-  ( A  =/=  (/)  ->  ran  ( ( R  \  ( A  X.  B ) )  i^i  ( A  X.  _V ) )  C_  ( _V  \  B ) )
65 disj2 3874 . . . . . . . . . . . 12  |-  ( ( ran  ( ( R 
\  ( A  X.  B ) )  i^i  ( A  X.  _V ) )  i^i  B
)  =  (/)  <->  ran  ( ( R  \  ( A  X.  B ) )  i^i  ( A  X.  _V ) )  C_  ( _V  \  B ) )
6664, 65sylibr 212 . . . . . . . . . . 11  |-  ( A  =/=  (/)  ->  ( ran  ( ( R  \ 
( A  X.  B
) )  i^i  ( A  X.  _V ) )  i^i  B )  =  (/) )
6752, 66syl 16 . . . . . . . . . 10  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  ( ran  ( ( R  \ 
( A  X.  B
) )  i^i  ( A  X.  _V ) )  i^i  B )  =  (/) )
68 ssdisj 3876 . . . . . . . . . 10  |-  ( ( ran  ( ( R 
\  ( A  X.  B ) )  i^i  ( C  X.  _V ) )  C_  ran  ( ( R  \ 
( A  X.  B
) )  i^i  ( A  X.  _V ) )  /\  ( ran  (
( R  \  ( A  X.  B ) )  i^i  ( A  X.  _V ) )  i^i  B
)  =  (/) )  -> 
( ran  ( ( R  \  ( A  X.  B ) )  i^i  ( C  X.  _V ) )  i^i  B
)  =  (/) )
6950, 67, 68syl2anc 661 . . . . . . . . 9  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  ( ran  ( ( R  \ 
( A  X.  B
) )  i^i  ( C  X.  _V ) )  i^i  B )  =  (/) )
7045, 69syl5eq 2520 . . . . . . . 8  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  (
( ( R  \ 
( A  X.  B
) ) " C
)  i^i  B )  =  (/) )
71 disj3 3871 . . . . . . . 8  |-  ( ( ( ( R  \ 
( A  X.  B
) ) " C
)  i^i  B )  =  (/)  <->  ( ( R 
\  ( A  X.  B ) ) " C )  =  ( ( ( R  \ 
( A  X.  B
) ) " C
)  \  B )
)
7270, 71sylib 196 . . . . . . 7  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  (
( R  \  ( A  X.  B ) )
" C )  =  ( ( ( R 
\  ( A  X.  B ) ) " C )  \  B
) )
7372eqcomd 2475 . . . . . 6  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  (
( ( R  \ 
( A  X.  B
) ) " C
)  \  B )  =  ( ( R 
\  ( A  X.  B ) ) " C ) )
7440, 73uneq12d 3659 . . . . 5  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  (
( ( ( R  i^i  ( A  X.  B ) ) " C )  \  B
)  u.  ( ( ( R  \  ( A  X.  B ) )
" C )  \  B ) )  =  ( (/)  u.  (
( R  \  ( A  X.  B ) )
" C ) ) )
7517, 74syl5eq 2520 . . . 4  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  (
( R " C
)  \  B )  =  ( (/)  u.  (
( R  \  ( A  X.  B ) )
" C ) ) )
76 uncom 3648 . . . . 5  |-  ( (/)  u.  ( ( R  \ 
( A  X.  B
) ) " C
) )  =  ( ( ( R  \ 
( A  X.  B
) ) " C
)  u.  (/) )
77 un0 3810 . . . . 5  |-  ( ( ( R  \  ( A  X.  B ) )
" C )  u.  (/) )  =  (
( R  \  ( A  X.  B ) )
" C )
7876, 77eqtr2i 2497 . . . 4  |-  ( ( R  \  ( A  X.  B ) )
" C )  =  ( (/)  u.  (
( R  \  ( A  X.  B ) )
" C ) )
7975, 78syl6reqr 2527 . . 3  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  (
( R  \  ( A  X.  B ) )
" C )  =  ( ( R " C )  \  B
) )
8079ancoms 453 . 2  |-  ( ( C  C_  A  /\  C  =/=  (/) )  ->  (
( R  \  ( A  X.  B ) )
" C )  =  ( ( R " C )  \  B
) )
8110, 80pm2.61dane 2785 1  |-  ( C 
C_  A  ->  (
( R  \  ( A  X.  B ) )
" C )  =  ( ( R " C )  \  B
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    = wceq 1379    =/= wne 2662   _Vcvv 3113    \ cdif 3473    u. cun 3474    i^i cin 3475    C_ wss 3476   (/)c0 3785   ifcif 3939    X. cxp 4997   ran crn 5000    |` cres 5001   "cima 5002
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 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  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 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-rel 5006  df-cnv 5007  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012
This theorem is referenced by: (None)
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