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Theorem resdif 5494
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 5452 . . . . . 6  |-  ( ( F  |`  A ) : A -onto-> C  ->  Fun  ( F  |`  A ) )
2 difss 3303 . . . . . . 7  |-  ( A 
\  B )  C_  A
3 fof 5451 . . . . . . . 8  |-  ( ( F  |`  A ) : A -onto-> C  ->  ( F  |`  A ) : A --> C )
4 fdm 5393 . . . . . . . 8  |-  ( ( F  |`  A ) : A --> C  ->  dom  ( F  |`  A )  =  A )
53, 4syl 15 . . . . . . 7  |-  ( ( F  |`  A ) : A -onto-> C  ->  dom  ( F  |`  A )  =  A )
62, 5syl5sseqr 3227 . . . . . 6  |-  ( ( F  |`  A ) : A -onto-> C  ->  ( A 
\  B )  C_  dom  ( F  |`  A ) )
7 fores 5460 . . . . . 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 4968 . . . . . . . 8  |-  ( ( F  |`  A )  |`  ( A  \  B
) )  =  ( F  |`  ( A  i^i  ( A  \  B
) ) )
10 indif 3411 . . . . . . . . 9  |-  ( A  i^i  ( A  \  B ) )  =  ( A  \  B
)
1110reseq2i 4952 . . . . . . . 8  |-  ( F  |`  ( A  i^i  ( A  \  B ) ) )  =  ( F  |`  ( A  \  B
) )
129, 11eqtri 2303 . . . . . . 7  |-  ( ( F  |`  A )  |`  ( A  \  B
) )  =  ( F  |`  ( A  \  B ) )
13 foeq1 5447 . . . . . . 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 4905 . . . . . . . 8  |-  ran  (
( F  |`  A )  |`  ( A  \  B
) )  =  ran  ( F  |`  ( A 
\  B ) )
16 df-ima 4702 . . . . . . . 8  |-  ( ( F  |`  A ) " ( A  \  B ) )  =  ran  ( ( F  |`  A )  |`  ( A  \  B ) )
17 df-ima 4702 . . . . . . . 8  |-  ( F
" ( A  \  B ) )  =  ran  ( F  |`  ( A  \  B ) )
1815, 16, 173eqtr4i 2313 . . . . . . 7  |-  ( ( F  |`  A ) " ( A  \  B ) )  =  ( F " ( A  \  B ) )
19 foeq3 5449 . . . . . . 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 5320 . . . 4  |-  ( Fun  `' F  ->  Fun  `' ( F  |`  ( A 
\  B ) ) )
24 dff1o3 5478 . . . . 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 4702 . . . . . . 7  |-  ( F
" A )  =  ran  ( F  |`  A )
29 forn 5454 . . . . . . 7  |-  ( ( F  |`  A ) : A -onto-> C  ->  ran  ( F  |`  A )  =  C )
3028, 29syl5eq 2327 . . . . . 6  |-  ( ( F  |`  A ) : A -onto-> C  ->  ( F
" A )  =  C )
31 df-ima 4702 . . . . . . 7  |-  ( F
" B )  =  ran  ( F  |`  B )
32 forn 5454 . . . . . . 7  |-  ( ( F  |`  B ) : B -onto-> D  ->  ran  ( F  |`  B )  =  D )
3331, 32syl5eq 2327 . . . . . 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 5327 . . . . . 6  |-  ( Fun  `' F  ->  ( F
" ( A  \  B ) )  =  ( ( F " A )  \  ( F " B ) ) )
36 difeq12 3289 . . . . . 6  |-  ( ( ( F " A
)  =  C  /\  ( F " B )  =  D )  -> 
( ( F " A )  \  ( F " B ) )  =  ( C  \  D ) )
3735, 36sylan9eq 2335 . . . . 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 5465 . . 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 1623    \ cdif 3149    i^i cin 3151    C_ wss 3152   `'ccnv 4688   dom cdm 4689   ran crn 4690    |` cres 4691   "cima 4692   Fun wfun 5249   -->wf 5251   -onto->wfo 5253   -1-1-onto->wf1o 5254
This theorem is referenced by:  resin  5495  canthp1lem2  8275  subfacp1lem3  23713  subfacp1lem5  23715
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262
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