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Theorem imadifxp 25944
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 5189 . . . 4  |-  ( ( R  \  ( A  X.  B ) )
" (/) )  =  (/)
2 imaeq2 5170 . . . 4  |-  ( C  =  (/)  ->  ( ( R  \  ( A  X.  B ) )
" C )  =  ( ( R  \ 
( A  X.  B
) ) " (/) ) )
3 imaeq2 5170 . . . . . . 7  |-  ( C  =  (/)  ->  ( R
" C )  =  ( R " (/) ) )
4 ima0 5189 . . . . . . 7  |-  ( R
" (/) )  =  (/)
53, 4syl6eq 2491 . . . . . 6  |-  ( C  =  (/)  ->  ( R
" C )  =  (/) )
65difeq1d 3478 . . . . 5  |-  ( C  =  (/)  ->  ( ( R " C ) 
\  B )  =  ( (/)  \  B ) )
7 0dif 3755 . . . . 5  |-  ( (/)  \  B )  =  (/)
86, 7syl6eq 2491 . . . 4  |-  ( C  =  (/)  ->  ( ( R " C ) 
\  B )  =  (/) )
91, 2, 83eqtr4a 2501 . . 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 3762 . . . . . . . . 9  |-  ( ( R  i^i  ( A  X.  B ) )  u.  ( R  \ 
( A  X.  B
) ) )  =  R
1211imaeq1i 5171 . . . . . . . 8  |-  ( ( ( R  i^i  ( A  X.  B ) )  u.  ( R  \ 
( A  X.  B
) ) ) " C )  =  ( R " C )
13 imaundir 5255 . . . . . . . 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 2465 . . . . . . 7  |-  ( R
" C )  =  ( ( ( R  i^i  ( A  X.  B ) ) " C )  u.  (
( R  \  ( A  X.  B ) )
" C ) )
1514difeq1i 3475 . . . . . 6  |-  ( ( R " C ) 
\  B )  =  ( ( ( ( R  i^i  ( A  X.  B ) )
" C )  u.  ( ( R  \ 
( A  X.  B
) ) " C
) )  \  B
)
16 difundir 3608 . . . . . 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 2463 . . . . 5  |-  ( ( R " C ) 
\  B )  =  ( ( ( ( R  i^i  ( A  X.  B ) )
" C )  \  B )  u.  (
( ( R  \ 
( A  X.  B
) ) " C
)  \  B )
)
18 inss2 3576 . . . . . . . . 9  |-  ( R  i^i  ( A  X.  B ) )  C_  ( A  X.  B
)
19 imass1 5208 . . . . . . . . 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 3496 . . . . . . . . 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 5285 . . . . . . . . . . 11  |-  ( ( A  X.  B )
" C )  =  if ( ( A  i^i  C )  =  (/) ,  (/) ,  B )
23 incom 3548 . . . . . . . . . . . . . . 15  |-  ( C  i^i  A )  =  ( A  i^i  C
)
24 df-ss 3347 . . . . . . . . . . . . . . . 16  |-  ( C 
C_  A  <->  ( C  i^i  A )  =  C )
2524biimpi 194 . . . . . . . . . . . . . . 15  |-  ( C 
C_  A  ->  ( C  i^i  A )  =  C )
2623, 25syl5eqr 2489 . . . . . . . . . . . . . 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 2631 . . . . . . . . . . . 12  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  ( A  i^i  C )  =/=  (/) )
30 df-ne 2613 . . . . . . . . . . . . 13  |-  ( ( A  i^i  C )  =/=  (/)  <->  -.  ( A  i^i  C )  =  (/) )
3130biimpi 194 . . . . . . . . . . . 12  |-  ( ( A  i^i  C )  =/=  (/)  ->  -.  ( A  i^i  C )  =  (/) )
32 iffalse 3804 . . . . . . . . . . . 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 2487 . . . . . . . . . 10  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  (
( A  X.  B
) " C )  =  B )
3534difeq1d 3478 . . . . . . . . 9  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  (
( ( A  X.  B ) " C
)  \  B )  =  ( B  \  B ) )
36 difid 3752 . . . . . . . . 9  |-  ( B 
\  B )  =  (/)
3735, 36syl6eq 2491 . . . . . . . 8  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  (
( ( A  X.  B ) " C
)  \  B )  =  (/) )
3821, 37syl5sseq 3409 . . . . . . 7  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  (
( ( R  i^i  ( A  X.  B
) ) " C
)  \  B )  C_  (/) )
39 ss0 3673 . . . . . . 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 4858 . . . . . . . . . . 11  |-  ( ( R  \  ( A  X.  B ) )
" C )  =  ran  ( ( R 
\  ( A  X.  B ) )  |`  C )
42 df-res 4857 . . . . . . . . . . . 12  |-  ( ( R  \  ( A  X.  B ) )  |`  C )  =  ( ( R  \  ( A  X.  B ) )  i^i  ( C  X.  _V ) )
4342rneqi 5071 . . . . . . . . . . 11  |-  ran  (
( R  \  ( A  X.  B ) )  |`  C )  =  ran  ( ( R  \ 
( A  X.  B
) )  i^i  ( C  X.  _V ) )
4441, 43eqtri 2463 . . . . . . . . . 10  |-  ( ( R  \  ( A  X.  B ) )
" C )  =  ran  ( ( R 
\  ( A  X.  B ) )  i^i  ( C  X.  _V ) )
4544ineq1i 3553 . . . . . . . . 9  |-  ( ( ( R  \  ( A  X.  B ) )
" C )  i^i 
B )  =  ( ran  ( ( R 
\  ( A  X.  B ) )  i^i  ( C  X.  _V ) )  i^i  B
)
46 xpss1 4953 . . . . . . . . . . . 12  |-  ( C 
C_  A  ->  ( C  X.  _V )  C_  ( A  X.  _V )
)
47 sslin 3581 . . . . . . . . . . . 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 5073 . . . . . . . . . . . 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 3675 . . . . . . . . . . . 12  |-  ( ( C  C_  A  /\  C  =/=  (/) )  ->  A  =/=  (/) )
5251ancoms 453 . . . . . . . . . . 11  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  A  =/=  (/) )
53 inss1 3575 . . . . . . . . . . . . . . . 16  |-  ( ( A  X.  _V )  i^i  R )  C_  ( A  X.  _V )
54 ssdif 3496 . . . . . . . . . . . . . . . 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 3548 . . . . . . . . . . . . . . . 16  |-  ( ( A  X.  _V )  i^i  ( R  \  ( A  X.  B ) ) )  =  ( ( R  \  ( A  X.  B ) )  i^i  ( A  X.  _V ) )
57 indif2 3598 . . . . . . . . . . . . . . . 16  |-  ( ( A  X.  _V )  i^i  ( R  \  ( A  X.  B ) ) )  =  ( ( ( A  X.  _V )  i^i  R )  \ 
( A  X.  B
) )
5856, 57eqtr3i 2465 . . . . . . . . . . . . . . 15  |-  ( ( R  \  ( A  X.  B ) )  i^i  ( A  X.  _V ) )  =  ( ( ( A  X.  _V )  i^i  R ) 
\  ( A  X.  B ) )
59 difxp2 5269 . . . . . . . . . . . . . . 15  |-  ( A  X.  ( _V  \  B ) )  =  ( ( A  X.  _V )  \  ( A  X.  B ) )
6055, 58, 593sstr4i 3400 . . . . . . . . . . . . . 14  |-  ( ( R  \  ( A  X.  B ) )  i^i  ( A  X.  _V ) )  C_  ( A  X.  ( _V  \  B ) )
61 rnss 5073 . . . . . . . . . . . . . 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 5273 . . . . . . . . . . . . 13  |-  ( A  =/=  (/)  ->  ran  ( A  X.  ( _V  \  B ) )  =  ( _V  \  B
) )
6462, 63sseqtrd 3397 . . . . . . . . . . . 12  |-  ( A  =/=  (/)  ->  ran  ( ( R  \  ( A  X.  B ) )  i^i  ( A  X.  _V ) )  C_  ( _V  \  B ) )
65 disj2 3731 . . . . . . . . . . . 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 3733 . . . . . . . . . 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 2487 . . . . . . . 8  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  (
( ( R  \ 
( A  X.  B
) ) " C
)  i^i  B )  =  (/) )
71 disj3 3728 . . . . . . . 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 2448 . . . . . 6  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  (
( ( R  \ 
( A  X.  B
) ) " C
)  \  B )  =  ( ( R 
\  ( A  X.  B ) ) " C ) )
7440, 73uneq12d 3516 . . . . 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 2487 . . . 4  |-  ( ( C  =/=  (/)  /\  C  C_  A )  ->  (
( R " C
)  \  B )  =  ( (/)  u.  (
( R  \  ( A  X.  B ) )
" C ) ) )
76 uncom 3505 . . . . 5  |-  ( (/)  u.  ( ( R  \ 
( A  X.  B
) ) " C
) )  =  ( ( ( R  \ 
( A  X.  B
) ) " C
)  u.  (/) )
77 un0 3667 . . . . 5  |-  ( ( ( R  \  ( A  X.  B ) )
" C )  u.  (/) )  =  (
( R  \  ( A  X.  B ) )
" C )
7876, 77eqtr2i 2464 . . . 4  |-  ( ( R  \  ( A  X.  B ) )
" C )  =  ( (/)  u.  (
( R  \  ( A  X.  B ) )
" C ) )
7975, 78syl6reqr 2494 . . 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 2694 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 1369    =/= wne 2611   _Vcvv 2977    \ cdif 3330    u. cun 3331    i^i cin 3332    C_ wss 3333   (/)c0 3642   ifcif 3796    X. cxp 4843   ran crn 4846    |` cres 4847   "cima 4848
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4418  ax-nul 4426  ax-pr 4536
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-rab 2729  df-v 2979  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-nul 3643  df-if 3797  df-sn 3883  df-pr 3885  df-op 3889  df-br 4298  df-opab 4356  df-xp 4851  df-rel 4852  df-cnv 4853  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858
This theorem is referenced by: (None)
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