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Theorem resdif 5510
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 5468 . . . . . 6  |-  ( ( F  |`  A ) : A -onto-> C  ->  Fun  ( F  |`  A ) )
2 difss 3316 . . . . . . 7  |-  ( A 
\  B )  C_  A
3 fof 5467 . . . . . . . 8  |-  ( ( F  |`  A ) : A -onto-> C  ->  ( F  |`  A ) : A --> C )
4 fdm 5409 . . . . . . . 8  |-  ( ( F  |`  A ) : A --> C  ->  dom  ( F  |`  A )  =  A )
53, 4syl 15 . . . . . . 7  |-  ( ( F  |`  A ) : A -onto-> C  ->  dom  ( F  |`  A )  =  A )
62, 5syl5sseqr 3240 . . . . . 6  |-  ( ( F  |`  A ) : A -onto-> C  ->  ( A 
\  B )  C_  dom  ( F  |`  A ) )
7 fores 5476 . . . . . 6  |-  ( ( Fun  ( F  |`  A )  /\  ( A  \  B )  C_  dom  ( F  |`  A ) )  ->  ( ( F  |`  A )  |`  ( A  \  B ) ) : ( A 
\  B ) -onto-> ( ( F  |`  A )
" ( A  \  B ) ) )
81, 6, 7syl2anc 642 . . . . 5  |-  ( ( F  |`  A ) : A -onto-> C  ->  ( ( F  |`  A )  |`  ( A  \  B
) ) : ( A  \  B )
-onto-> ( ( F  |`  A ) " ( A  \  B ) ) )
9 resres 4984 . . . . . . . 8  |-  ( ( F  |`  A )  |`  ( A  \  B
) )  =  ( F  |`  ( A  i^i  ( A  \  B
) ) )
10 indif 3424 . . . . . . . . 9  |-  ( A  i^i  ( A  \  B ) )  =  ( A  \  B
)
1110reseq2i 4968 . . . . . . . 8  |-  ( F  |`  ( A  i^i  ( A  \  B ) ) )  =  ( F  |`  ( A  \  B
) )
129, 11eqtri 2316 . . . . . . 7  |-  ( ( F  |`  A )  |`  ( A  \  B
) )  =  ( F  |`  ( A  \  B ) )
13 foeq1 5463 . . . . . . 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 8 . . . . . 6  |-  ( ( ( F  |`  A )  |`  ( A  \  B
) ) : ( A  \  B )
-onto-> ( ( F  |`  A ) " ( A  \  B ) )  <-> 
( F  |`  ( A  \  B ) ) : ( A  \  B ) -onto-> ( ( F  |`  A ) " ( A  \  B ) ) )
1512rneqi 4921 . . . . . . . 8  |-  ran  (
( F  |`  A )  |`  ( A  \  B
) )  =  ran  ( F  |`  ( A 
\  B ) )
16 df-ima 4718 . . . . . . . 8  |-  ( ( F  |`  A ) " ( A  \  B ) )  =  ran  ( ( F  |`  A )  |`  ( A  \  B ) )
17 df-ima 4718 . . . . . . . 8  |-  ( F
" ( A  \  B ) )  =  ran  ( F  |`  ( A  \  B ) )
1815, 16, 173eqtr4i 2326 . . . . . . 7  |-  ( ( F  |`  A ) " ( A  \  B ) )  =  ( F " ( A  \  B ) )
19 foeq3 5465 . . . . . . 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 8 . . . . . 6  |-  ( ( F  |`  ( A  \  B ) ) : ( A  \  B
) -onto-> ( ( F  |`  A ) " ( A  \  B ) )  <-> 
( F  |`  ( A  \  B ) ) : ( A  \  B ) -onto-> ( F
" ( A  \  B ) ) )
2114, 20bitri 240 . . . . 5  |-  ( ( ( F  |`  A )  |`  ( A  \  B
) ) : ( A  \  B )
-onto-> ( ( F  |`  A ) " ( A  \  B ) )  <-> 
( F  |`  ( A  \  B ) ) : ( A  \  B ) -onto-> ( F
" ( A  \  B ) ) )
228, 21sylib 188 . . . 4  |-  ( ( F  |`  A ) : A -onto-> C  ->  ( F  |`  ( A  \  B
) ) : ( A  \  B )
-onto-> ( F " ( A  \  B ) ) )
23 funres11 5336 . . . 4  |-  ( Fun  `' F  ->  Fun  `' ( F  |`  ( A 
\  B ) ) )
24 dff1o3 5494 . . . . 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 197 . . . 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 464 . . 3  |-  ( ( Fun  `' F  /\  ( F  |`  A ) : A -onto-> C )  ->  ( F  |`  ( A  \  B ) ) : ( A 
\  B ) -1-1-onto-> ( F
" ( A  \  B ) ) )
27263adant3 975 . 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 4718 . . . . . . 7  |-  ( F
" A )  =  ran  ( F  |`  A )
29 forn 5470 . . . . . . 7  |-  ( ( F  |`  A ) : A -onto-> C  ->  ran  ( F  |`  A )  =  C )
3028, 29syl5eq 2340 . . . . . 6  |-  ( ( F  |`  A ) : A -onto-> C  ->  ( F
" A )  =  C )
31 df-ima 4718 . . . . . . 7  |-  ( F
" B )  =  ran  ( F  |`  B )
32 forn 5470 . . . . . . 7  |-  ( ( F  |`  B ) : B -onto-> D  ->  ran  ( F  |`  B )  =  D )
3331, 32syl5eq 2340 . . . . . 6  |-  ( ( F  |`  B ) : B -onto-> D  ->  ( F
" B )  =  D )
3430, 33anim12i 549 . . . . 5  |-  ( ( ( F  |`  A ) : A -onto-> C  /\  ( F  |`  B ) : B -onto-> D )  ->  ( ( F
" A )  =  C  /\  ( F
" B )  =  D ) )
35 imadif 5343 . . . . . 6  |-  ( Fun  `' F  ->  ( F
" ( A  \  B ) )  =  ( ( F " A )  \  ( F " B ) ) )
36 difeq12 3302 . . . . . 6  |-  ( ( ( F " A
)  =  C  /\  ( F " B )  =  D )  -> 
( ( F " A )  \  ( F " B ) )  =  ( C  \  D ) )
3735, 36sylan9eq 2348 . . . . 5  |-  ( ( Fun  `' F  /\  ( ( F " A )  =  C  /\  ( F " B )  =  D ) )  ->  ( F " ( A  \  B ) )  =  ( C  \  D
) )
3834, 37sylan2 460 . . . 4  |-  ( ( Fun  `' F  /\  ( ( F  |`  A ) : A -onto-> C  /\  ( F  |`  B ) : B -onto-> D ) )  -> 
( F " ( A  \  B ) )  =  ( C  \  D ) )
39383impb 1147 . . 3  |-  ( ( Fun  `' F  /\  ( F  |`  A ) : A -onto-> C  /\  ( F  |`  B ) : B -onto-> D )  ->  ( F "
( A  \  B
) )  =  ( C  \  D ) )
40 f1oeq3 5481 . . 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 15 . 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 201 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    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1632    \ cdif 3162    i^i cin 3164    C_ wss 3165   `'ccnv 4704   dom cdm 4705   ran crn 4706    |` cres 4707   "cima 4708   Fun wfun 5265   -->wf 5267   -onto->wfo 5269   -1-1-onto->wf1o 5270
This theorem is referenced by:  resin  5511  canthp1lem2  8291  subfacp1lem3  23728  subfacp1lem5  23730
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278
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