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Theorem resdif 6614
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 6570 . . . . . 6 ((𝐹𝐴):𝐴onto𝐶 → Fun (𝐹𝐴))
2 difss 4062 . . . . . . 7 (𝐴𝐵) ⊆ 𝐴
3 fof 6569 . . . . . . . 8 ((𝐹𝐴):𝐴onto𝐶 → (𝐹𝐴):𝐴𝐶)
43fdmd 6501 . . . . . . 7 ((𝐹𝐴):𝐴onto𝐶 → dom (𝐹𝐴) = 𝐴)
52, 4sseqtrrid 3971 . . . . . 6 ((𝐹𝐴):𝐴onto𝐶 → (𝐴𝐵) ⊆ dom (𝐹𝐴))
6 fores 6579 . . . . . 6 ((Fun (𝐹𝐴) ∧ (𝐴𝐵) ⊆ dom (𝐹𝐴)) → ((𝐹𝐴) ↾ (𝐴𝐵)):(𝐴𝐵)–onto→((𝐹𝐴) “ (𝐴𝐵)))
71, 5, 6syl2anc 587 . . . . 5 ((𝐹𝐴):𝐴onto𝐶 → ((𝐹𝐴) ↾ (𝐴𝐵)):(𝐴𝐵)–onto→((𝐹𝐴) “ (𝐴𝐵)))
8 resres 5835 . . . . . . . 8 ((𝐹𝐴) ↾ (𝐴𝐵)) = (𝐹 ↾ (𝐴 ∩ (𝐴𝐵)))
9 indif 4199 . . . . . . . . 9 (𝐴 ∩ (𝐴𝐵)) = (𝐴𝐵)
109reseq2i 5819 . . . . . . . 8 (𝐹 ↾ (𝐴 ∩ (𝐴𝐵))) = (𝐹 ↾ (𝐴𝐵))
118, 10eqtri 2824 . . . . . . 7 ((𝐹𝐴) ↾ (𝐴𝐵)) = (𝐹 ↾ (𝐴𝐵))
12 foeq1 6565 . . . . . . 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 5775 . . . . . . . 8 ran ((𝐹𝐴) ↾ (𝐴𝐵)) = ran (𝐹 ↾ (𝐴𝐵))
15 df-ima 5536 . . . . . . . 8 ((𝐹𝐴) “ (𝐴𝐵)) = ran ((𝐹𝐴) ↾ (𝐴𝐵))
16 df-ima 5536 . . . . . . . 8 (𝐹 “ (𝐴𝐵)) = ran (𝐹 ↾ (𝐴𝐵))
1714, 15, 163eqtr4i 2834 . . . . . . 7 ((𝐹𝐴) “ (𝐴𝐵)) = (𝐹 “ (𝐴𝐵))
18 foeq3 6567 . . . . . . 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 278 . . . . 5 (((𝐹𝐴) ↾ (𝐴𝐵)):(𝐴𝐵)–onto→((𝐹𝐴) “ (𝐴𝐵)) ↔ (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–onto→(𝐹 “ (𝐴𝐵)))
217, 20sylib 221 . . . 4 ((𝐹𝐴):𝐴onto𝐶 → (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–onto→(𝐹 “ (𝐴𝐵)))
22 funres11 6405 . . . 4 (Fun 𝐹 → Fun (𝐹 ↾ (𝐴𝐵)))
23 dff1o3 6600 . . . . 5 ((𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–1-1-onto→(𝐹 “ (𝐴𝐵)) ↔ ((𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–onto→(𝐹 “ (𝐴𝐵)) ∧ Fun (𝐹 ↾ (𝐴𝐵))))
2423biimpri 231 . . . 4 (((𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–onto→(𝐹 “ (𝐴𝐵)) ∧ Fun (𝐹 ↾ (𝐴𝐵))) → (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–1-1-onto→(𝐹 “ (𝐴𝐵)))
2521, 22, 24syl2anr 599 . . 3 ((Fun 𝐹 ∧ (𝐹𝐴):𝐴onto𝐶) → (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–1-1-onto→(𝐹 “ (𝐴𝐵)))
26253adant3 1129 . 2 ((Fun 𝐹 ∧ (𝐹𝐴):𝐴onto𝐶 ∧ (𝐹𝐵):𝐵onto𝐷) → (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–1-1-onto→(𝐹 “ (𝐴𝐵)))
27 df-ima 5536 . . . . . . 7 (𝐹𝐴) = ran (𝐹𝐴)
28 forn 6572 . . . . . . 7 ((𝐹𝐴):𝐴onto𝐶 → ran (𝐹𝐴) = 𝐶)
2927, 28syl5eq 2848 . . . . . 6 ((𝐹𝐴):𝐴onto𝐶 → (𝐹𝐴) = 𝐶)
30 df-ima 5536 . . . . . . 7 (𝐹𝐵) = ran (𝐹𝐵)
31 forn 6572 . . . . . . 7 ((𝐹𝐵):𝐵onto𝐷 → ran (𝐹𝐵) = 𝐷)
3230, 31syl5eq 2848 . . . . . 6 ((𝐹𝐵):𝐵onto𝐷 → (𝐹𝐵) = 𝐷)
3329, 32anim12i 615 . . . . 5 (((𝐹𝐴):𝐴onto𝐶 ∧ (𝐹𝐵):𝐵onto𝐷) → ((𝐹𝐴) = 𝐶 ∧ (𝐹𝐵) = 𝐷))
34 imadif 6412 . . . . . 6 (Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∖ (𝐹𝐵)))
35 difeq12 4048 . . . . . 6 (((𝐹𝐴) = 𝐶 ∧ (𝐹𝐵) = 𝐷) → ((𝐹𝐴) ∖ (𝐹𝐵)) = (𝐶𝐷))
3634, 35sylan9eq 2856 . . . . 5 ((Fun 𝐹 ∧ ((𝐹𝐴) = 𝐶 ∧ (𝐹𝐵) = 𝐷)) → (𝐹 “ (𝐴𝐵)) = (𝐶𝐷))
3733, 36sylan2 595 . . . 4 ((Fun 𝐹 ∧ ((𝐹𝐴):𝐴onto𝐶 ∧ (𝐹𝐵):𝐵onto𝐷)) → (𝐹 “ (𝐴𝐵)) = (𝐶𝐷))
38373impb 1112 . . 3 ((Fun 𝐹 ∧ (𝐹𝐴):𝐴onto𝐶 ∧ (𝐹𝐵):𝐵onto𝐷) → (𝐹 “ (𝐴𝐵)) = (𝐶𝐷))
3938f1oeq3d 6591 . 2 ((Fun 𝐹 ∧ (𝐹𝐴):𝐴onto𝐶 ∧ (𝐹𝐵):𝐵onto𝐷) → ((𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–1-1-onto→(𝐹 “ (𝐴𝐵)) ↔ (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–1-1-onto→(𝐶𝐷)))
4026, 39mpbid 235 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 209  wa 399  w3a 1084   = wceq 1538  cdif 3881  cin 3883  wss 3884  ccnv 5522  dom cdm 5523  ran crn 5524  cres 5525  cima 5526  Fun wfun 6322  ontowfo 6326  1-1-ontowf1o 6327
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335
This theorem is referenced by:  resin  6615  canthp1lem2  10068  symgcom  30781  cycpmconjvlem  30837  subfacp1lem3  32543  subfacp1lem5  32545
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