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Theorem resdif 5502
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 5458 . . . . . 6 ((𝐹𝐴):𝐴onto𝐶 → Fun (𝐹𝐴))
2 difss 3276 . . . . . . 7 (𝐴𝐵) ⊆ 𝐴
3 fof 5457 . . . . . . . 8 ((𝐹𝐴):𝐴onto𝐶 → (𝐹𝐴):𝐴𝐶)
4 fdm 5390 . . . . . . . 8 ((𝐹𝐴):𝐴𝐶 → dom (𝐹𝐴) = 𝐴)
53, 4syl 14 . . . . . . 7 ((𝐹𝐴):𝐴onto𝐶 → dom (𝐹𝐴) = 𝐴)
62, 5sseqtrrid 3221 . . . . . 6 ((𝐹𝐴):𝐴onto𝐶 → (𝐴𝐵) ⊆ dom (𝐹𝐴))
7 fores 5466 . . . . . 6 ((Fun (𝐹𝐴) ∧ (𝐴𝐵) ⊆ dom (𝐹𝐴)) → ((𝐹𝐴) ↾ (𝐴𝐵)):(𝐴𝐵)–onto→((𝐹𝐴) “ (𝐴𝐵)))
81, 6, 7syl2anc 411 . . . . 5 ((𝐹𝐴):𝐴onto𝐶 → ((𝐹𝐴) ↾ (𝐴𝐵)):(𝐴𝐵)–onto→((𝐹𝐴) “ (𝐴𝐵)))
9 resres 4937 . . . . . . . 8 ((𝐹𝐴) ↾ (𝐴𝐵)) = (𝐹 ↾ (𝐴 ∩ (𝐴𝐵)))
10 indif 3393 . . . . . . . . 9 (𝐴 ∩ (𝐴𝐵)) = (𝐴𝐵)
1110reseq2i 4922 . . . . . . . 8 (𝐹 ↾ (𝐴 ∩ (𝐴𝐵))) = (𝐹 ↾ (𝐴𝐵))
129, 11eqtri 2210 . . . . . . 7 ((𝐹𝐴) ↾ (𝐴𝐵)) = (𝐹 ↾ (𝐴𝐵))
13 foeq1 5453 . . . . . . 7 (((𝐹𝐴) ↾ (𝐴𝐵)) = (𝐹 ↾ (𝐴𝐵)) → (((𝐹𝐴) ↾ (𝐴𝐵)):(𝐴𝐵)–onto→((𝐹𝐴) “ (𝐴𝐵)) ↔ (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–onto→((𝐹𝐴) “ (𝐴𝐵))))
1412, 13ax-mp 5 . . . . . 6 (((𝐹𝐴) ↾ (𝐴𝐵)):(𝐴𝐵)–onto→((𝐹𝐴) “ (𝐴𝐵)) ↔ (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–onto→((𝐹𝐴) “ (𝐴𝐵)))
1512rneqi 4873 . . . . . . . 8 ran ((𝐹𝐴) ↾ (𝐴𝐵)) = ran (𝐹 ↾ (𝐴𝐵))
16 df-ima 4657 . . . . . . . 8 ((𝐹𝐴) “ (𝐴𝐵)) = ran ((𝐹𝐴) ↾ (𝐴𝐵))
17 df-ima 4657 . . . . . . . 8 (𝐹 “ (𝐴𝐵)) = ran (𝐹 ↾ (𝐴𝐵))
1815, 16, 173eqtr4i 2220 . . . . . . 7 ((𝐹𝐴) “ (𝐴𝐵)) = (𝐹 “ (𝐴𝐵))
19 foeq3 5455 . . . . . . 7 (((𝐹𝐴) “ (𝐴𝐵)) = (𝐹 “ (𝐴𝐵)) → ((𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–onto→((𝐹𝐴) “ (𝐴𝐵)) ↔ (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–onto→(𝐹 “ (𝐴𝐵))))
2018, 19ax-mp 5 . . . . . 6 ((𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–onto→((𝐹𝐴) “ (𝐴𝐵)) ↔ (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–onto→(𝐹 “ (𝐴𝐵)))
2114, 20bitri 184 . . . . 5 (((𝐹𝐴) ↾ (𝐴𝐵)):(𝐴𝐵)–onto→((𝐹𝐴) “ (𝐴𝐵)) ↔ (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–onto→(𝐹 “ (𝐴𝐵)))
228, 21sylib 122 . . . 4 ((𝐹𝐴):𝐴onto𝐶 → (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–onto→(𝐹 “ (𝐴𝐵)))
23 funres11 5307 . . . 4 (Fun 𝐹 → Fun (𝐹 ↾ (𝐴𝐵)))
24 dff1o3 5486 . . . . 5 ((𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–1-1-onto→(𝐹 “ (𝐴𝐵)) ↔ ((𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–onto→(𝐹 “ (𝐴𝐵)) ∧ Fun (𝐹 ↾ (𝐴𝐵))))
2524biimpri 133 . . . 4 (((𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–onto→(𝐹 “ (𝐴𝐵)) ∧ Fun (𝐹 ↾ (𝐴𝐵))) → (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–1-1-onto→(𝐹 “ (𝐴𝐵)))
2622, 23, 25syl2anr 290 . . 3 ((Fun 𝐹 ∧ (𝐹𝐴):𝐴onto𝐶) → (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–1-1-onto→(𝐹 “ (𝐴𝐵)))
27263adant3 1019 . 2 ((Fun 𝐹 ∧ (𝐹𝐴):𝐴onto𝐶 ∧ (𝐹𝐵):𝐵onto𝐷) → (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–1-1-onto→(𝐹 “ (𝐴𝐵)))
28 df-ima 4657 . . . . . . 7 (𝐹𝐴) = ran (𝐹𝐴)
29 forn 5460 . . . . . . 7 ((𝐹𝐴):𝐴onto𝐶 → ran (𝐹𝐴) = 𝐶)
3028, 29eqtrid 2234 . . . . . 6 ((𝐹𝐴):𝐴onto𝐶 → (𝐹𝐴) = 𝐶)
31 df-ima 4657 . . . . . . 7 (𝐹𝐵) = ran (𝐹𝐵)
32 forn 5460 . . . . . . 7 ((𝐹𝐵):𝐵onto𝐷 → ran (𝐹𝐵) = 𝐷)
3331, 32eqtrid 2234 . . . . . 6 ((𝐹𝐵):𝐵onto𝐷 → (𝐹𝐵) = 𝐷)
3430, 33anim12i 338 . . . . 5 (((𝐹𝐴):𝐴onto𝐶 ∧ (𝐹𝐵):𝐵onto𝐷) → ((𝐹𝐴) = 𝐶 ∧ (𝐹𝐵) = 𝐷))
35 imadif 5315 . . . . . 6 (Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∖ (𝐹𝐵)))
36 difeq12 3263 . . . . . 6 (((𝐹𝐴) = 𝐶 ∧ (𝐹𝐵) = 𝐷) → ((𝐹𝐴) ∖ (𝐹𝐵)) = (𝐶𝐷))
3735, 36sylan9eq 2242 . . . . 5 ((Fun 𝐹 ∧ ((𝐹𝐴) = 𝐶 ∧ (𝐹𝐵) = 𝐷)) → (𝐹 “ (𝐴𝐵)) = (𝐶𝐷))
3834, 37sylan2 286 . . . 4 ((Fun 𝐹 ∧ ((𝐹𝐴):𝐴onto𝐶 ∧ (𝐹𝐵):𝐵onto𝐷)) → (𝐹 “ (𝐴𝐵)) = (𝐶𝐷))
39383impb 1201 . . 3 ((Fun 𝐹 ∧ (𝐹𝐴):𝐴onto𝐶 ∧ (𝐹𝐵):𝐵onto𝐷) → (𝐹 “ (𝐴𝐵)) = (𝐶𝐷))
40 f1oeq3 5470 . . 3 ((𝐹 “ (𝐴𝐵)) = (𝐶𝐷) → ((𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–1-1-onto→(𝐹 “ (𝐴𝐵)) ↔ (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–1-1-onto→(𝐶𝐷)))
4139, 40syl 14 . 2 ((Fun 𝐹 ∧ (𝐹𝐴):𝐴onto𝐶 ∧ (𝐹𝐵):𝐵onto𝐷) → ((𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–1-1-onto→(𝐹 “ (𝐴𝐵)) ↔ (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–1-1-onto→(𝐶𝐷)))
4227, 41mpbid 147 1 ((Fun 𝐹 ∧ (𝐹𝐴):𝐴onto𝐶 ∧ (𝐹𝐵):𝐵onto𝐷) → (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–1-1-onto→(𝐶𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  w3a 980   = wceq 1364  cdif 3141  cin 3143  wss 3144  ccnv 4643  dom cdm 4644  ran crn 4645  cres 4646  cima 4647  Fun wfun 5229  wf 5231  ontowfo 5233  1-1-ontowf1o 5234
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-rab 2477  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-opab 4080  df-id 4311  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242
This theorem is referenced by:  dif1en  6908
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