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

Theorem resdif 6825
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 6777 . . . . . 6 ((𝐹𝐴):𝐴onto𝐶 → Fun (𝐹𝐴))
2 difss 4111 . . . . . . 7 (𝐴𝐵) ⊆ 𝐴
3 fof 6776 . . . . . . . 8 ((𝐹𝐴):𝐴onto𝐶 → (𝐹𝐴):𝐴𝐶)
43fdmd 6699 . . . . . . 7 ((𝐹𝐴):𝐴onto𝐶 → dom (𝐹𝐴) = 𝐴)
52, 4sseqtrrid 4015 . . . . . 6 ((𝐹𝐴):𝐴onto𝐶 → (𝐴𝐵) ⊆ dom (𝐹𝐴))
6 fores 6786 . . . . . 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 5970 . . . . . . . 8 ((𝐹𝐴) ↾ (𝐴𝐵)) = (𝐹 ↾ (𝐴 ∩ (𝐴𝐵)))
9 indif 4249 . . . . . . . . 9 (𝐴 ∩ (𝐴𝐵)) = (𝐴𝐵)
109reseq2i 5954 . . . . . . . 8 (𝐹 ↾ (𝐴 ∩ (𝐴𝐵))) = (𝐹 ↾ (𝐴𝐵))
118, 10eqtri 2759 . . . . . . 7 ((𝐹𝐴) ↾ (𝐴𝐵)) = (𝐹 ↾ (𝐴𝐵))
12 foeq1 6772 . . . . . . 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 5912 . . . . . . . 8 ran ((𝐹𝐴) ↾ (𝐴𝐵)) = ran (𝐹 ↾ (𝐴𝐵))
15 df-ima 5666 . . . . . . . 8 ((𝐹𝐴) “ (𝐴𝐵)) = ran ((𝐹𝐴) ↾ (𝐴𝐵))
16 df-ima 5666 . . . . . . . 8 (𝐹 “ (𝐴𝐵)) = ran (𝐹 ↾ (𝐴𝐵))
1714, 15, 163eqtr4i 2769 . . . . . . 7 ((𝐹𝐴) “ (𝐴𝐵)) = (𝐹 “ (𝐴𝐵))
18 foeq3 6774 . . . . . . 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 274 . . . . 5 (((𝐹𝐴) ↾ (𝐴𝐵)):(𝐴𝐵)–onto→((𝐹𝐴) “ (𝐴𝐵)) ↔ (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–onto→(𝐹 “ (𝐴𝐵)))
217, 20sylib 217 . . . 4 ((𝐹𝐴):𝐴onto𝐶 → (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–onto→(𝐹 “ (𝐴𝐵)))
22 funres11 6598 . . . 4 (Fun 𝐹 → Fun (𝐹 ↾ (𝐴𝐵)))
23 dff1o3 6810 . . . . 5 ((𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–1-1-onto→(𝐹 “ (𝐴𝐵)) ↔ ((𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–onto→(𝐹 “ (𝐴𝐵)) ∧ Fun (𝐹 ↾ (𝐴𝐵))))
2423biimpri 227 . . . 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 5666 . . . . . . 7 (𝐹𝐴) = ran (𝐹𝐴)
28 forn 6779 . . . . . . 7 ((𝐹𝐴):𝐴onto𝐶 → ran (𝐹𝐴) = 𝐶)
2927, 28eqtrid 2783 . . . . . 6 ((𝐹𝐴):𝐴onto𝐶 → (𝐹𝐴) = 𝐶)
30 df-ima 5666 . . . . . . 7 (𝐹𝐵) = ran (𝐹𝐵)
31 forn 6779 . . . . . . 7 ((𝐹𝐵):𝐵onto𝐷 → ran (𝐹𝐵) = 𝐷)
3230, 31eqtrid 2783 . . . . . 6 ((𝐹𝐵):𝐵onto𝐷 → (𝐹𝐵) = 𝐷)
3329, 32anim12i 613 . . . . 5 (((𝐹𝐴):𝐴onto𝐶 ∧ (𝐹𝐵):𝐵onto𝐷) → ((𝐹𝐴) = 𝐶 ∧ (𝐹𝐵) = 𝐷))
34 imadif 6605 . . . . . 6 (Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∖ (𝐹𝐵)))
35 difeq12 4097 . . . . . 6 (((𝐹𝐴) = 𝐶 ∧ (𝐹𝐵) = 𝐷) → ((𝐹𝐴) ∖ (𝐹𝐵)) = (𝐶𝐷))
3634, 35sylan9eq 2791 . . . . 5 ((Fun 𝐹 ∧ ((𝐹𝐴) = 𝐶 ∧ (𝐹𝐵) = 𝐷)) → (𝐹 “ (𝐴𝐵)) = (𝐶𝐷))
3733, 36sylan2 593 . . . 4 ((Fun 𝐹 ∧ ((𝐹𝐴):𝐴onto𝐶 ∧ (𝐹𝐵):𝐵onto𝐷)) → (𝐹 “ (𝐴𝐵)) = (𝐶𝐷))
38373impb 1115 . . 3 ((Fun 𝐹 ∧ (𝐹𝐴):𝐴onto𝐶 ∧ (𝐹𝐵):𝐵onto𝐷) → (𝐹 “ (𝐴𝐵)) = (𝐶𝐷))
3938f1oeq3d 6801 . 2 ((Fun 𝐹 ∧ (𝐹𝐴):𝐴onto𝐶 ∧ (𝐹𝐵):𝐵onto𝐷) → ((𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–1-1-onto→(𝐹 “ (𝐴𝐵)) ↔ (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–1-1-onto→(𝐶𝐷)))
4026, 39mpbid 231 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 205  wa 396  w3a 1087   = wceq 1541  cdif 3925  cin 3927  wss 3928  ccnv 5652  dom cdm 5653  ran crn 5654  cres 5655  cima 5656  Fun wfun 6510  ontowfo 6514  1-1-ontowf1o 6515
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2702  ax-sep 5276  ax-nul 5283  ax-pr 5404
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3419  df-v 3461  df-dif 3931  df-un 3933  df-in 3935  df-ss 3945  df-nul 4303  df-if 4507  df-sn 4607  df-pr 4609  df-op 4613  df-br 5126  df-opab 5188  df-id 5551  df-xp 5659  df-rel 5660  df-cnv 5661  df-co 5662  df-dm 5663  df-rn 5664  df-res 5665  df-ima 5666  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523
This theorem is referenced by:  resin  6826  dif1enlem  9122  dif1enlemOLD  9123  canthp1lem2  10613  symgcom  32038  cycpmconjvlem  32094  subfacp1lem3  33897  subfacp1lem5  33899
  Copyright terms: Public domain W3C validator