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Theorem resdif 5666
Description: The restriction of a one-to-one onto function to a difference maps onto the difference of the images. (Contributed by Paul Chapman, 11-Apr-2009.)
Assertion
Ref Expression
resdif  |-  ( ( Fun  `' F  /\  ( F  |`  A ) : A -onto-> C  /\  ( F  |`  B ) : B -onto-> D )  ->  ( F  |`  ( A  \  B ) ) : ( A 
\  B ) -1-1-onto-> ( C 
\  D ) )

Proof of Theorem resdif
StepHypRef Expression
1 fofun 5626 . . . . . 6  |-  ( ( F  |`  A ) : A -onto-> C  ->  Fun  ( F  |`  A ) )
2 difss 3488 . . . . . . 7  |-  ( A 
\  B )  C_  A
3 fof 5625 . . . . . . . 8  |-  ( ( F  |`  A ) : A -onto-> C  ->  ( F  |`  A ) : A --> C )
4 fdm 5568 . . . . . . . 8  |-  ( ( F  |`  A ) : A --> C  ->  dom  ( F  |`  A )  =  A )
53, 4syl 16 . . . . . . 7  |-  ( ( F  |`  A ) : A -onto-> C  ->  dom  ( F  |`  A )  =  A )
62, 5syl5sseqr 3410 . . . . . 6  |-  ( ( F  |`  A ) : A -onto-> C  ->  ( A 
\  B )  C_  dom  ( F  |`  A ) )
7 fores 5634 . . . . . 6  |-  ( ( Fun  ( F  |`  A )  /\  ( A  \  B )  C_  dom  ( F  |`  A ) )  ->  ( ( F  |`  A )  |`  ( A  \  B ) ) : ( A 
\  B ) -onto-> ( ( F  |`  A )
" ( A  \  B ) ) )
81, 6, 7syl2anc 661 . . . . 5  |-  ( ( F  |`  A ) : A -onto-> C  ->  ( ( F  |`  A )  |`  ( A  \  B
) ) : ( A  \  B )
-onto-> ( ( F  |`  A ) " ( A  \  B ) ) )
9 resres 5128 . . . . . . . 8  |-  ( ( F  |`  A )  |`  ( A  \  B
) )  =  ( F  |`  ( A  i^i  ( A  \  B
) ) )
10 indif 3597 . . . . . . . . 9  |-  ( A  i^i  ( A  \  B ) )  =  ( A  \  B
)
1110reseq2i 5112 . . . . . . . 8  |-  ( F  |`  ( A  i^i  ( A  \  B ) ) )  =  ( F  |`  ( A  \  B
) )
129, 11eqtri 2463 . . . . . . 7  |-  ( ( F  |`  A )  |`  ( A  \  B
) )  =  ( F  |`  ( A  \  B ) )
13 foeq1 5621 . . . . . . 7  |-  ( ( ( F  |`  A )  |`  ( A  \  B
) )  =  ( F  |`  ( A  \  B ) )  -> 
( ( ( F  |`  A )  |`  ( A  \  B ) ) : ( A  \  B ) -onto-> ( ( F  |`  A ) " ( A  \  B ) )  <->  ( F  |`  ( A  \  B
) ) : ( A  \  B )
-onto-> ( ( F  |`  A ) " ( A  \  B ) ) ) )
1412, 13ax-mp 5 . . . . . 6  |-  ( ( ( F  |`  A )  |`  ( A  \  B
) ) : ( A  \  B )
-onto-> ( ( F  |`  A ) " ( A  \  B ) )  <-> 
( F  |`  ( A  \  B ) ) : ( A  \  B ) -onto-> ( ( F  |`  A ) " ( A  \  B ) ) )
1512rneqi 5071 . . . . . . . 8  |-  ran  (
( F  |`  A )  |`  ( A  \  B
) )  =  ran  ( F  |`  ( A 
\  B ) )
16 df-ima 4858 . . . . . . . 8  |-  ( ( F  |`  A ) " ( A  \  B ) )  =  ran  ( ( F  |`  A )  |`  ( A  \  B ) )
17 df-ima 4858 . . . . . . . 8  |-  ( F
" ( A  \  B ) )  =  ran  ( F  |`  ( A  \  B ) )
1815, 16, 173eqtr4i 2473 . . . . . . 7  |-  ( ( F  |`  A ) " ( A  \  B ) )  =  ( F " ( A  \  B ) )
19 foeq3 5623 . . . . . . 7  |-  ( ( ( F  |`  A )
" ( A  \  B ) )  =  ( F " ( A  \  B ) )  ->  ( ( F  |`  ( A  \  B
) ) : ( A  \  B )
-onto-> ( ( F  |`  A ) " ( A  \  B ) )  <-> 
( F  |`  ( A  \  B ) ) : ( A  \  B ) -onto-> ( F
" ( A  \  B ) ) ) )
2018, 19ax-mp 5 . . . . . 6  |-  ( ( F  |`  ( A  \  B ) ) : ( A  \  B
) -onto-> ( ( F  |`  A ) " ( A  \  B ) )  <-> 
( F  |`  ( A  \  B ) ) : ( A  \  B ) -onto-> ( F
" ( A  \  B ) ) )
2114, 20bitri 249 . . . . 5  |-  ( ( ( F  |`  A )  |`  ( A  \  B
) ) : ( A  \  B )
-onto-> ( ( F  |`  A ) " ( A  \  B ) )  <-> 
( F  |`  ( A  \  B ) ) : ( A  \  B ) -onto-> ( F
" ( A  \  B ) ) )
228, 21sylib 196 . . . 4  |-  ( ( F  |`  A ) : A -onto-> C  ->  ( F  |`  ( A  \  B
) ) : ( A  \  B )
-onto-> ( F " ( A  \  B ) ) )
23 funres11 5491 . . . 4  |-  ( Fun  `' F  ->  Fun  `' ( F  |`  ( A 
\  B ) ) )
24 dff1o3 5652 . . . . 5  |-  ( ( F  |`  ( A  \  B ) ) : ( A  \  B
)
-1-1-onto-> ( F " ( A 
\  B ) )  <-> 
( ( F  |`  ( A  \  B ) ) : ( A 
\  B ) -onto-> ( F " ( A 
\  B ) )  /\  Fun  `' ( F  |`  ( A  \  B ) ) ) )
2524biimpri 206 . . . 4  |-  ( ( ( F  |`  ( A  \  B ) ) : ( A  \  B ) -onto-> ( F
" ( A  \  B ) )  /\  Fun  `' ( F  |`  ( A  \  B ) ) )  ->  ( F  |`  ( A  \  B ) ) : ( A  \  B
)
-1-1-onto-> ( F " ( A 
\  B ) ) )
2622, 23, 25syl2anr 478 . . 3  |-  ( ( Fun  `' F  /\  ( F  |`  A ) : A -onto-> C )  ->  ( F  |`  ( A  \  B ) ) : ( A 
\  B ) -1-1-onto-> ( F
" ( A  \  B ) ) )
27263adant3 1008 . 2  |-  ( ( Fun  `' F  /\  ( F  |`  A ) : A -onto-> C  /\  ( F  |`  B ) : B -onto-> D )  ->  ( F  |`  ( A  \  B ) ) : ( A 
\  B ) -1-1-onto-> ( F
" ( A  \  B ) ) )
28 df-ima 4858 . . . . . . 7  |-  ( F
" A )  =  ran  ( F  |`  A )
29 forn 5628 . . . . . . 7  |-  ( ( F  |`  A ) : A -onto-> C  ->  ran  ( F  |`  A )  =  C )
3028, 29syl5eq 2487 . . . . . 6  |-  ( ( F  |`  A ) : A -onto-> C  ->  ( F
" A )  =  C )
31 df-ima 4858 . . . . . . 7  |-  ( F
" B )  =  ran  ( F  |`  B )
32 forn 5628 . . . . . . 7  |-  ( ( F  |`  B ) : B -onto-> D  ->  ran  ( F  |`  B )  =  D )
3331, 32syl5eq 2487 . . . . . 6  |-  ( ( F  |`  B ) : B -onto-> D  ->  ( F
" B )  =  D )
3430, 33anim12i 566 . . . . 5  |-  ( ( ( F  |`  A ) : A -onto-> C  /\  ( F  |`  B ) : B -onto-> D )  ->  ( ( F
" A )  =  C  /\  ( F
" B )  =  D ) )
35 imadif 5498 . . . . . 6  |-  ( Fun  `' F  ->  ( F
" ( A  \  B ) )  =  ( ( F " A )  \  ( F " B ) ) )
36 difeq12 3474 . . . . . 6  |-  ( ( ( F " A
)  =  C  /\  ( F " B )  =  D )  -> 
( ( F " A )  \  ( F " B ) )  =  ( C  \  D ) )
3735, 36sylan9eq 2495 . . . . 5  |-  ( ( Fun  `' F  /\  ( ( F " A )  =  C  /\  ( F " B )  =  D ) )  ->  ( F " ( A  \  B ) )  =  ( C  \  D
) )
3834, 37sylan2 474 . . . 4  |-  ( ( Fun  `' F  /\  ( ( F  |`  A ) : A -onto-> C  /\  ( F  |`  B ) : B -onto-> D ) )  -> 
( F " ( A  \  B ) )  =  ( C  \  D ) )
39383impb 1183 . . 3  |-  ( ( Fun  `' F  /\  ( F  |`  A ) : A -onto-> C  /\  ( F  |`  B ) : B -onto-> D )  ->  ( F "
( A  \  B
) )  =  ( C  \  D ) )
40 f1oeq3 5639 . . 3  |-  ( ( F " ( A 
\  B ) )  =  ( C  \  D )  ->  (
( F  |`  ( A  \  B ) ) : ( A  \  B ) -1-1-onto-> ( F " ( A  \  B ) )  <-> 
( F  |`  ( A  \  B ) ) : ( A  \  B ) -1-1-onto-> ( C  \  D
) ) )
4139, 40syl 16 . 2  |-  ( ( Fun  `' F  /\  ( F  |`  A ) : A -onto-> C  /\  ( F  |`  B ) : B -onto-> D )  ->  ( ( F  |`  ( A  \  B
) ) : ( A  \  B ) -1-1-onto-> ( F " ( A 
\  B ) )  <-> 
( F  |`  ( A  \  B ) ) : ( A  \  B ) -1-1-onto-> ( C  \  D
) ) )
4227, 41mpbid 210 1  |-  ( ( Fun  `' F  /\  ( F  |`  A ) : A -onto-> C  /\  ( F  |`  B ) : B -onto-> D )  ->  ( F  |`  ( A  \  B ) ) : ( A 
\  B ) -1-1-onto-> ( C 
\  D ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    = wceq 1369    \ cdif 3330    i^i cin 3332    C_ wss 3333   `'ccnv 4844   dom cdm 4845   ran crn 4846    |` cres 4847   "cima 4848   Fun wfun 5417   -->wf 5419   -onto->wfo 5421   -1-1-onto->wf1o 5422
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-id 4641  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430
This theorem is referenced by:  resin  5667  canthp1lem2  8825  subfacp1lem3  27075  subfacp1lem5  27077
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