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Theorem resdif 5259
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 5218 . . . . . 6  |-  ( ( F  |`  A ) : A -onto-> C  ->  Fun  ( F  |`  A ) )
2 difss 3124 . . . . . . 7  |-  ( A 
\  B )  C_  A
3 fof 5217 . . . . . . . 8  |-  ( ( F  |`  A ) : A -onto-> C  ->  ( F  |`  A ) : A --> C )
4 fdm 5152 . . . . . . . 8  |-  ( ( F  |`  A ) : A --> C  ->  dom  ( F  |`  A )  =  A )
53, 4syl 14 . . . . . . 7  |-  ( ( F  |`  A ) : A -onto-> C  ->  dom  ( F  |`  A )  =  A )
62, 5syl5sseqr 3073 . . . . . 6  |-  ( ( F  |`  A ) : A -onto-> C  ->  ( A 
\  B )  C_  dom  ( F  |`  A ) )
7 fores 5226 . . . . . 6  |-  ( ( Fun  ( F  |`  A )  /\  ( A  \  B )  C_  dom  ( F  |`  A ) )  ->  ( ( F  |`  A )  |`  ( A  \  B ) ) : ( A 
\  B ) -onto-> ( ( F  |`  A )
" ( A  \  B ) ) )
81, 6, 7syl2anc 403 . . . . 5  |-  ( ( F  |`  A ) : A -onto-> C  ->  ( ( F  |`  A )  |`  ( A  \  B
) ) : ( A  \  B )
-onto-> ( ( F  |`  A ) " ( A  \  B ) ) )
9 resres 4713 . . . . . . . 8  |-  ( ( F  |`  A )  |`  ( A  \  B
) )  =  ( F  |`  ( A  i^i  ( A  \  B
) ) )
10 indif 3240 . . . . . . . . 9  |-  ( A  i^i  ( A  \  B ) )  =  ( A  \  B
)
1110reseq2i 4698 . . . . . . . 8  |-  ( F  |`  ( A  i^i  ( A  \  B ) ) )  =  ( F  |`  ( A  \  B
) )
129, 11eqtri 2108 . . . . . . 7  |-  ( ( F  |`  A )  |`  ( A  \  B
) )  =  ( F  |`  ( A  \  B ) )
13 foeq1 5213 . . . . . . 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 4651 . . . . . . . 8  |-  ran  (
( F  |`  A )  |`  ( A  \  B
) )  =  ran  ( F  |`  ( A 
\  B ) )
16 df-ima 4441 . . . . . . . 8  |-  ( ( F  |`  A ) " ( A  \  B ) )  =  ran  ( ( F  |`  A )  |`  ( A  \  B ) )
17 df-ima 4441 . . . . . . . 8  |-  ( F
" ( A  \  B ) )  =  ran  ( F  |`  ( A  \  B ) )
1815, 16, 173eqtr4i 2118 . . . . . . 7  |-  ( ( F  |`  A ) " ( A  \  B ) )  =  ( F " ( A  \  B ) )
19 foeq3 5215 . . . . . . 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 182 . . . . 5  |-  ( ( ( F  |`  A )  |`  ( A  \  B
) ) : ( A  \  B )
-onto-> ( ( F  |`  A ) " ( A  \  B ) )  <-> 
( F  |`  ( A  \  B ) ) : ( A  \  B ) -onto-> ( F
" ( A  \  B ) ) )
228, 21sylib 120 . . . 4  |-  ( ( F  |`  A ) : A -onto-> C  ->  ( F  |`  ( A  \  B
) ) : ( A  \  B )
-onto-> ( F " ( A  \  B ) ) )
23 funres11 5072 . . . 4  |-  ( Fun  `' F  ->  Fun  `' ( F  |`  ( A 
\  B ) ) )
24 dff1o3 5243 . . . . 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 131 . . . 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 284 . . 3  |-  ( ( Fun  `' F  /\  ( F  |`  A ) : A -onto-> C )  ->  ( F  |`  ( A  \  B ) ) : ( A 
\  B ) -1-1-onto-> ( F
" ( A  \  B ) ) )
27263adant3 963 . 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 4441 . . . . . . 7  |-  ( F
" A )  =  ran  ( F  |`  A )
29 forn 5220 . . . . . . 7  |-  ( ( F  |`  A ) : A -onto-> C  ->  ran  ( F  |`  A )  =  C )
3028, 29syl5eq 2132 . . . . . 6  |-  ( ( F  |`  A ) : A -onto-> C  ->  ( F
" A )  =  C )
31 df-ima 4441 . . . . . . 7  |-  ( F
" B )  =  ran  ( F  |`  B )
32 forn 5220 . . . . . . 7  |-  ( ( F  |`  B ) : B -onto-> D  ->  ran  ( F  |`  B )  =  D )
3331, 32syl5eq 2132 . . . . . 6  |-  ( ( F  |`  B ) : B -onto-> D  ->  ( F
" B )  =  D )
3430, 33anim12i 331 . . . . 5  |-  ( ( ( F  |`  A ) : A -onto-> C  /\  ( F  |`  B ) : B -onto-> D )  ->  ( ( F
" A )  =  C  /\  ( F
" B )  =  D ) )
35 imadif 5080 . . . . . 6  |-  ( Fun  `' F  ->  ( F
" ( A  \  B ) )  =  ( ( F " A )  \  ( F " B ) ) )
36 difeq12 3111 . . . . . 6  |-  ( ( ( F " A
)  =  C  /\  ( F " B )  =  D )  -> 
( ( F " A )  \  ( F " B ) )  =  ( C  \  D ) )
3735, 36sylan9eq 2140 . . . . 5  |-  ( ( Fun  `' F  /\  ( ( F " A )  =  C  /\  ( F " B )  =  D ) )  ->  ( F " ( A  \  B ) )  =  ( C  \  D
) )
3834, 37sylan2 280 . . . 4  |-  ( ( Fun  `' F  /\  ( ( F  |`  A ) : A -onto-> C  /\  ( F  |`  B ) : B -onto-> D ) )  -> 
( F " ( A  \  B ) )  =  ( C  \  D ) )
39383impb 1139 . . 3  |-  ( ( Fun  `' F  /\  ( F  |`  A ) : A -onto-> C  /\  ( F  |`  B ) : B -onto-> D )  ->  ( F "
( A  \  B
) )  =  ( C  \  D ) )
40 f1oeq3 5230 . . 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 145 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 102    <-> wb 103    /\ w3a 924    = wceq 1289    \ cdif 2994    i^i cin 2996    C_ wss 2997   `'ccnv 4427   dom cdm 4428   ran crn 4429    |` cres 4430   "cima 4431   Fun wfun 4996   -->wf 4998   -onto->wfo 5000   -1-1-onto->wf1o 5001
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-rab 2368  df-v 2621  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-id 4111  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009
This theorem is referenced by:  dif1en  6575
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