MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  resdif Structured version   Visualization version   GIF version

Theorem resdif 6868
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 ∧ (𝐹𝐴):𝐴onto𝐶 ∧ (𝐹𝐵):𝐵onto𝐷) → (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–1-1-onto→(𝐶𝐷))

Proof of Theorem resdif
StepHypRef Expression
1 fofun 6820 . . . . . 6 ((𝐹𝐴):𝐴onto𝐶 → Fun (𝐹𝐴))
2 difss 4135 . . . . . . 7 (𝐴𝐵) ⊆ 𝐴
3 fof 6819 . . . . . . . 8 ((𝐹𝐴):𝐴onto𝐶 → (𝐹𝐴):𝐴𝐶)
43fdmd 6745 . . . . . . 7 ((𝐹𝐴):𝐴onto𝐶 → dom (𝐹𝐴) = 𝐴)
52, 4sseqtrrid 4026 . . . . . 6 ((𝐹𝐴):𝐴onto𝐶 → (𝐴𝐵) ⊆ dom (𝐹𝐴))
6 fores 6829 . . . . . 6 ((Fun (𝐹𝐴) ∧ (𝐴𝐵) ⊆ dom (𝐹𝐴)) → ((𝐹𝐴) ↾ (𝐴𝐵)):(𝐴𝐵)–onto→((𝐹𝐴) “ (𝐴𝐵)))
71, 5, 6syl2anc 584 . . . . 5 ((𝐹𝐴):𝐴onto𝐶 → ((𝐹𝐴) ↾ (𝐴𝐵)):(𝐴𝐵)–onto→((𝐹𝐴) “ (𝐴𝐵)))
8 resres 6009 . . . . . . . 8 ((𝐹𝐴) ↾ (𝐴𝐵)) = (𝐹 ↾ (𝐴 ∩ (𝐴𝐵)))
9 indif 4279 . . . . . . . . 9 (𝐴 ∩ (𝐴𝐵)) = (𝐴𝐵)
109reseq2i 5993 . . . . . . . 8 (𝐹 ↾ (𝐴 ∩ (𝐴𝐵))) = (𝐹 ↾ (𝐴𝐵))
118, 10eqtri 2764 . . . . . . 7 ((𝐹𝐴) ↾ (𝐴𝐵)) = (𝐹 ↾ (𝐴𝐵))
12 foeq1 6815 . . . . . . 7 (((𝐹𝐴) ↾ (𝐴𝐵)) = (𝐹 ↾ (𝐴𝐵)) → (((𝐹𝐴) ↾ (𝐴𝐵)):(𝐴𝐵)–onto→((𝐹𝐴) “ (𝐴𝐵)) ↔ (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–onto→((𝐹𝐴) “ (𝐴𝐵))))
1311, 12ax-mp 5 . . . . . 6 (((𝐹𝐴) ↾ (𝐴𝐵)):(𝐴𝐵)–onto→((𝐹𝐴) “ (𝐴𝐵)) ↔ (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–onto→((𝐹𝐴) “ (𝐴𝐵)))
1411rneqi 5947 . . . . . . . 8 ran ((𝐹𝐴) ↾ (𝐴𝐵)) = ran (𝐹 ↾ (𝐴𝐵))
15 df-ima 5697 . . . . . . . 8 ((𝐹𝐴) “ (𝐴𝐵)) = ran ((𝐹𝐴) ↾ (𝐴𝐵))
16 df-ima 5697 . . . . . . . 8 (𝐹 “ (𝐴𝐵)) = ran (𝐹 ↾ (𝐴𝐵))
1714, 15, 163eqtr4i 2774 . . . . . . 7 ((𝐹𝐴) “ (𝐴𝐵)) = (𝐹 “ (𝐴𝐵))
18 foeq3 6817 . . . . . . 7 (((𝐹𝐴) “ (𝐴𝐵)) = (𝐹 “ (𝐴𝐵)) → ((𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–onto→((𝐹𝐴) “ (𝐴𝐵)) ↔ (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–onto→(𝐹 “ (𝐴𝐵))))
1917, 18ax-mp 5 . . . . . 6 ((𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–onto→((𝐹𝐴) “ (𝐴𝐵)) ↔ (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–onto→(𝐹 “ (𝐴𝐵)))
2013, 19bitri 275 . . . . 5 (((𝐹𝐴) ↾ (𝐴𝐵)):(𝐴𝐵)–onto→((𝐹𝐴) “ (𝐴𝐵)) ↔ (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–onto→(𝐹 “ (𝐴𝐵)))
217, 20sylib 218 . . . 4 ((𝐹𝐴):𝐴onto𝐶 → (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–onto→(𝐹 “ (𝐴𝐵)))
22 funres11 6642 . . . 4 (Fun 𝐹 → Fun (𝐹 ↾ (𝐴𝐵)))
23 dff1o3 6853 . . . . 5 ((𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–1-1-onto→(𝐹 “ (𝐴𝐵)) ↔ ((𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–onto→(𝐹 “ (𝐴𝐵)) ∧ Fun (𝐹 ↾ (𝐴𝐵))))
2423biimpri 228 . . . 4 (((𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–onto→(𝐹 “ (𝐴𝐵)) ∧ Fun (𝐹 ↾ (𝐴𝐵))) → (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–1-1-onto→(𝐹 “ (𝐴𝐵)))
2521, 22, 24syl2anr 597 . . 3 ((Fun 𝐹 ∧ (𝐹𝐴):𝐴onto𝐶) → (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–1-1-onto→(𝐹 “ (𝐴𝐵)))
26253adant3 1132 . 2 ((Fun 𝐹 ∧ (𝐹𝐴):𝐴onto𝐶 ∧ (𝐹𝐵):𝐵onto𝐷) → (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–1-1-onto→(𝐹 “ (𝐴𝐵)))
27 df-ima 5697 . . . . . . 7 (𝐹𝐴) = ran (𝐹𝐴)
28 forn 6822 . . . . . . 7 ((𝐹𝐴):𝐴onto𝐶 → ran (𝐹𝐴) = 𝐶)
2927, 28eqtrid 2788 . . . . . 6 ((𝐹𝐴):𝐴onto𝐶 → (𝐹𝐴) = 𝐶)
30 df-ima 5697 . . . . . . 7 (𝐹𝐵) = ran (𝐹𝐵)
31 forn 6822 . . . . . . 7 ((𝐹𝐵):𝐵onto𝐷 → ran (𝐹𝐵) = 𝐷)
3230, 31eqtrid 2788 . . . . . 6 ((𝐹𝐵):𝐵onto𝐷 → (𝐹𝐵) = 𝐷)
3329, 32anim12i 613 . . . . 5 (((𝐹𝐴):𝐴onto𝐶 ∧ (𝐹𝐵):𝐵onto𝐷) → ((𝐹𝐴) = 𝐶 ∧ (𝐹𝐵) = 𝐷))
34 imadif 6649 . . . . . 6 (Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∖ (𝐹𝐵)))
35 difeq12 4120 . . . . . 6 (((𝐹𝐴) = 𝐶 ∧ (𝐹𝐵) = 𝐷) → ((𝐹𝐴) ∖ (𝐹𝐵)) = (𝐶𝐷))
3634, 35sylan9eq 2796 . . . . 5 ((Fun 𝐹 ∧ ((𝐹𝐴) = 𝐶 ∧ (𝐹𝐵) = 𝐷)) → (𝐹 “ (𝐴𝐵)) = (𝐶𝐷))
3733, 36sylan2 593 . . . 4 ((Fun 𝐹 ∧ ((𝐹𝐴):𝐴onto𝐶 ∧ (𝐹𝐵):𝐵onto𝐷)) → (𝐹 “ (𝐴𝐵)) = (𝐶𝐷))
38373impb 1114 . . 3 ((Fun 𝐹 ∧ (𝐹𝐴):𝐴onto𝐶 ∧ (𝐹𝐵):𝐵onto𝐷) → (𝐹 “ (𝐴𝐵)) = (𝐶𝐷))
3938f1oeq3d 6844 . 2 ((Fun 𝐹 ∧ (𝐹𝐴):𝐴onto𝐶 ∧ (𝐹𝐵):𝐵onto𝐷) → ((𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–1-1-onto→(𝐹 “ (𝐴𝐵)) ↔ (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–1-1-onto→(𝐶𝐷)))
4026, 39mpbid 232 1 ((Fun 𝐹 ∧ (𝐹𝐴):𝐴onto𝐶 ∧ (𝐹𝐵):𝐵onto𝐷) → (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–1-1-onto→(𝐶𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1539  cdif 3947  cin 3949  wss 3950  ccnv 5683  dom cdm 5684  ran crn 5685  cres 5686  cima 5687  Fun wfun 6554  ontowfo 6558  1-1-ontowf1o 6559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-id 5577  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567
This theorem is referenced by:  resin  6869  dif1enlem  9197  dif1enlemOLD  9198  canthp1lem2  10694  symgcom  33104  cycpmconjvlem  33162  subfacp1lem3  35188  subfacp1lem5  35190
  Copyright terms: Public domain W3C validator