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Theorem resdif 5176
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 5135 . . . . . 6  |-  ( ( F  |`  A ) : A -onto-> C  ->  Fun  ( F  |`  A ) )
2 difss 3098 . . . . . . 7  |-  ( A 
\  B )  C_  A
3 fof 5134 . . . . . . . 8  |-  ( ( F  |`  A ) : A -onto-> C  ->  ( F  |`  A ) : A --> C )
4 fdm 5078 . . . . . . . 8  |-  ( ( F  |`  A ) : A --> C  ->  dom  ( F  |`  A )  =  A )
53, 4syl 14 . . . . . . 7  |-  ( ( F  |`  A ) : A -onto-> C  ->  dom  ( F  |`  A )  =  A )
62, 5syl5sseqr 3022 . . . . . 6  |-  ( ( F  |`  A ) : A -onto-> C  ->  ( A 
\  B )  C_  dom  ( F  |`  A ) )
7 fores 5143 . . . . . 6  |-  ( ( Fun  ( F  |`  A )  /\  ( A  \  B )  C_  dom  ( F  |`  A ) )  ->  ( ( F  |`  A )  |`  ( A  \  B ) ) : ( A 
\  B ) -onto-> ( ( F  |`  A )
" ( A  \  B ) ) )
81, 6, 7syl2anc 397 . . . . 5  |-  ( ( F  |`  A ) : A -onto-> C  ->  ( ( F  |`  A )  |`  ( A  \  B
) ) : ( A  \  B )
-onto-> ( ( F  |`  A ) " ( A  \  B ) ) )
9 resres 4652 . . . . . . . 8  |-  ( ( F  |`  A )  |`  ( A  \  B
) )  =  ( F  |`  ( A  i^i  ( A  \  B
) ) )
10 indif 3208 . . . . . . . . 9  |-  ( A  i^i  ( A  \  B ) )  =  ( A  \  B
)
1110reseq2i 4637 . . . . . . . 8  |-  ( F  |`  ( A  i^i  ( A  \  B ) ) )  =  ( F  |`  ( A  \  B
) )
129, 11eqtri 2076 . . . . . . 7  |-  ( ( F  |`  A )  |`  ( A  \  B
) )  =  ( F  |`  ( A  \  B ) )
13 foeq1 5130 . . . . . . 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 7 . . . . . 6  |-  ( ( ( F  |`  A )  |`  ( A  \  B
) ) : ( A  \  B )
-onto-> ( ( F  |`  A ) " ( A  \  B ) )  <-> 
( F  |`  ( A  \  B ) ) : ( A  \  B ) -onto-> ( ( F  |`  A ) " ( A  \  B ) ) )
1512rneqi 4590 . . . . . . . 8  |-  ran  (
( F  |`  A )  |`  ( A  \  B
) )  =  ran  ( F  |`  ( A 
\  B ) )
16 df-ima 4386 . . . . . . . 8  |-  ( ( F  |`  A ) " ( A  \  B ) )  =  ran  ( ( F  |`  A )  |`  ( A  \  B ) )
17 df-ima 4386 . . . . . . . 8  |-  ( F
" ( A  \  B ) )  =  ran  ( F  |`  ( A  \  B ) )
1815, 16, 173eqtr4i 2086 . . . . . . 7  |-  ( ( F  |`  A ) " ( A  \  B ) )  =  ( F " ( A  \  B ) )
19 foeq3 5132 . . . . . . 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 7 . . . . . 6  |-  ( ( F  |`  ( A  \  B ) ) : ( A  \  B
) -onto-> ( ( F  |`  A ) " ( A  \  B ) )  <-> 
( F  |`  ( A  \  B ) ) : ( A  \  B ) -onto-> ( F
" ( A  \  B ) ) )
2114, 20bitri 177 . . . . 5  |-  ( ( ( F  |`  A )  |`  ( A  \  B
) ) : ( A  \  B )
-onto-> ( ( F  |`  A ) " ( A  \  B ) )  <-> 
( F  |`  ( A  \  B ) ) : ( A  \  B ) -onto-> ( F
" ( A  \  B ) ) )
228, 21sylib 131 . . . 4  |-  ( ( F  |`  A ) : A -onto-> C  ->  ( F  |`  ( A  \  B
) ) : ( A  \  B )
-onto-> ( F " ( A  \  B ) ) )
23 funres11 4999 . . . 4  |-  ( Fun  `' F  ->  Fun  `' ( F  |`  ( A 
\  B ) ) )
24 dff1o3 5160 . . . . 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 128 . . . 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 278 . . 3  |-  ( ( Fun  `' F  /\  ( F  |`  A ) : A -onto-> C )  ->  ( F  |`  ( A  \  B ) ) : ( A 
\  B ) -1-1-onto-> ( F
" ( A  \  B ) ) )
27263adant3 935 . 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 4386 . . . . . . 7  |-  ( F
" A )  =  ran  ( F  |`  A )
29 forn 5137 . . . . . . 7  |-  ( ( F  |`  A ) : A -onto-> C  ->  ran  ( F  |`  A )  =  C )
3028, 29syl5eq 2100 . . . . . 6  |-  ( ( F  |`  A ) : A -onto-> C  ->  ( F
" A )  =  C )
31 df-ima 4386 . . . . . . 7  |-  ( F
" B )  =  ran  ( F  |`  B )
32 forn 5137 . . . . . . 7  |-  ( ( F  |`  B ) : B -onto-> D  ->  ran  ( F  |`  B )  =  D )
3331, 32syl5eq 2100 . . . . . 6  |-  ( ( F  |`  B ) : B -onto-> D  ->  ( F
" B )  =  D )
3430, 33anim12i 325 . . . . 5  |-  ( ( ( F  |`  A ) : A -onto-> C  /\  ( F  |`  B ) : B -onto-> D )  ->  ( ( F
" A )  =  C  /\  ( F
" B )  =  D ) )
35 imadif 5007 . . . . . 6  |-  ( Fun  `' F  ->  ( F
" ( A  \  B ) )  =  ( ( F " A )  \  ( F " B ) ) )
36 difeq12 3085 . . . . . 6  |-  ( ( ( F " A
)  =  C  /\  ( F " B )  =  D )  -> 
( ( F " A )  \  ( F " B ) )  =  ( C  \  D ) )
3735, 36sylan9eq 2108 . . . . 5  |-  ( ( Fun  `' F  /\  ( ( F " A )  =  C  /\  ( F " B )  =  D ) )  ->  ( F " ( A  \  B ) )  =  ( C  \  D
) )
3834, 37sylan2 274 . . . 4  |-  ( ( Fun  `' F  /\  ( ( F  |`  A ) : A -onto-> C  /\  ( F  |`  B ) : B -onto-> D ) )  -> 
( F " ( A  \  B ) )  =  ( C  \  D ) )
39383impb 1111 . . 3  |-  ( ( Fun  `' F  /\  ( F  |`  A ) : A -onto-> C  /\  ( F  |`  B ) : B -onto-> D )  ->  ( F "
( A  \  B
) )  =  ( C  \  D ) )
40 f1oeq3 5147 . . 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 14 . 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 139 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 set class
Syntax hints:    -> wi 4    /\ wa 101    <-> wb 102    /\ w3a 896    = wceq 1259    \ cdif 2942    i^i cin 2944    C_ wss 2945   `'ccnv 4372   dom cdm 4373   ran crn 4374    |` cres 4375   "cima 4376   Fun wfun 4924   -->wf 4926   -onto->wfo 4928   -1-1-onto->wf1o 4929
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-id 4058  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937
This theorem is referenced by:  dif1en  6368
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