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Theorem resdif 5396
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 5353 . . . . . 6 ((𝐹𝐴):𝐴onto𝐶 → Fun (𝐹𝐴))
2 difss 3206 . . . . . . 7 (𝐴𝐵) ⊆ 𝐴
3 fof 5352 . . . . . . . 8 ((𝐹𝐴):𝐴onto𝐶 → (𝐹𝐴):𝐴𝐶)
4 fdm 5285 . . . . . . . 8 ((𝐹𝐴):𝐴𝐶 → dom (𝐹𝐴) = 𝐴)
53, 4syl 14 . . . . . . 7 ((𝐹𝐴):𝐴onto𝐶 → dom (𝐹𝐴) = 𝐴)
62, 5sseqtrrid 3152 . . . . . 6 ((𝐹𝐴):𝐴onto𝐶 → (𝐴𝐵) ⊆ dom (𝐹𝐴))
7 fores 5361 . . . . . 6 ((Fun (𝐹𝐴) ∧ (𝐴𝐵) ⊆ dom (𝐹𝐴)) → ((𝐹𝐴) ↾ (𝐴𝐵)):(𝐴𝐵)–onto→((𝐹𝐴) “ (𝐴𝐵)))
81, 6, 7syl2anc 409 . . . . 5 ((𝐹𝐴):𝐴onto𝐶 → ((𝐹𝐴) ↾ (𝐴𝐵)):(𝐴𝐵)–onto→((𝐹𝐴) “ (𝐴𝐵)))
9 resres 4838 . . . . . . . 8 ((𝐹𝐴) ↾ (𝐴𝐵)) = (𝐹 ↾ (𝐴 ∩ (𝐴𝐵)))
10 indif 3323 . . . . . . . . 9 (𝐴 ∩ (𝐴𝐵)) = (𝐴𝐵)
1110reseq2i 4823 . . . . . . . 8 (𝐹 ↾ (𝐴 ∩ (𝐴𝐵))) = (𝐹 ↾ (𝐴𝐵))
129, 11eqtri 2161 . . . . . . 7 ((𝐹𝐴) ↾ (𝐴𝐵)) = (𝐹 ↾ (𝐴𝐵))
13 foeq1 5348 . . . . . . 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 4774 . . . . . . . 8 ran ((𝐹𝐴) ↾ (𝐴𝐵)) = ran (𝐹 ↾ (𝐴𝐵))
16 df-ima 4559 . . . . . . . 8 ((𝐹𝐴) “ (𝐴𝐵)) = ran ((𝐹𝐴) ↾ (𝐴𝐵))
17 df-ima 4559 . . . . . . . 8 (𝐹 “ (𝐴𝐵)) = ran (𝐹 ↾ (𝐴𝐵))
1815, 16, 173eqtr4i 2171 . . . . . . 7 ((𝐹𝐴) “ (𝐴𝐵)) = (𝐹 “ (𝐴𝐵))
19 foeq3 5350 . . . . . . 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 183 . . . . 5 (((𝐹𝐴) ↾ (𝐴𝐵)):(𝐴𝐵)–onto→((𝐹𝐴) “ (𝐴𝐵)) ↔ (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–onto→(𝐹 “ (𝐴𝐵)))
228, 21sylib 121 . . . 4 ((𝐹𝐴):𝐴onto𝐶 → (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–onto→(𝐹 “ (𝐴𝐵)))
23 funres11 5202 . . . 4 (Fun 𝐹 → Fun (𝐹 ↾ (𝐴𝐵)))
24 dff1o3 5380 . . . . 5 ((𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–1-1-onto→(𝐹 “ (𝐴𝐵)) ↔ ((𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–onto→(𝐹 “ (𝐴𝐵)) ∧ Fun (𝐹 ↾ (𝐴𝐵))))
2524biimpri 132 . . . 4 (((𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–onto→(𝐹 “ (𝐴𝐵)) ∧ Fun (𝐹 ↾ (𝐴𝐵))) → (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–1-1-onto→(𝐹 “ (𝐴𝐵)))
2622, 23, 25syl2anr 288 . . 3 ((Fun 𝐹 ∧ (𝐹𝐴):𝐴onto𝐶) → (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–1-1-onto→(𝐹 “ (𝐴𝐵)))
27263adant3 1002 . 2 ((Fun 𝐹 ∧ (𝐹𝐴):𝐴onto𝐶 ∧ (𝐹𝐵):𝐵onto𝐷) → (𝐹 ↾ (𝐴𝐵)):(𝐴𝐵)–1-1-onto→(𝐹 “ (𝐴𝐵)))
28 df-ima 4559 . . . . . . 7 (𝐹𝐴) = ran (𝐹𝐴)
29 forn 5355 . . . . . . 7 ((𝐹𝐴):𝐴onto𝐶 → ran (𝐹𝐴) = 𝐶)
3028, 29syl5eq 2185 . . . . . 6 ((𝐹𝐴):𝐴onto𝐶 → (𝐹𝐴) = 𝐶)
31 df-ima 4559 . . . . . . 7 (𝐹𝐵) = ran (𝐹𝐵)
32 forn 5355 . . . . . . 7 ((𝐹𝐵):𝐵onto𝐷 → ran (𝐹𝐵) = 𝐷)
3331, 32syl5eq 2185 . . . . . 6 ((𝐹𝐵):𝐵onto𝐷 → (𝐹𝐵) = 𝐷)
3430, 33anim12i 336 . . . . 5 (((𝐹𝐴):𝐴onto𝐶 ∧ (𝐹𝐵):𝐵onto𝐷) → ((𝐹𝐴) = 𝐶 ∧ (𝐹𝐵) = 𝐷))
35 imadif 5210 . . . . . 6 (Fun 𝐹 → (𝐹 “ (𝐴𝐵)) = ((𝐹𝐴) ∖ (𝐹𝐵)))
36 difeq12 3193 . . . . . 6 (((𝐹𝐴) = 𝐶 ∧ (𝐹𝐵) = 𝐷) → ((𝐹𝐴) ∖ (𝐹𝐵)) = (𝐶𝐷))
3735, 36sylan9eq 2193 . . . . 5 ((Fun 𝐹 ∧ ((𝐹𝐴) = 𝐶 ∧ (𝐹𝐵) = 𝐷)) → (𝐹 “ (𝐴𝐵)) = (𝐶𝐷))
3834, 37sylan2 284 . . . 4 ((Fun 𝐹 ∧ ((𝐹𝐴):𝐴onto𝐶 ∧ (𝐹𝐵):𝐵onto𝐷)) → (𝐹 “ (𝐴𝐵)) = (𝐶𝐷))
39383impb 1178 . . 3 ((Fun 𝐹 ∧ (𝐹𝐴):𝐴onto𝐶 ∧ (𝐹𝐵):𝐵onto𝐷) → (𝐹 “ (𝐴𝐵)) = (𝐶𝐷))
40 f1oeq3 5365 . . 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 146 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 103  wb 104  w3a 963   = wceq 1332  cdif 3072  cin 3074  wss 3075  ccnv 4545  dom cdm 4546  ran crn 4547  cres 4548  cima 4549  Fun wfun 5124  wf 5126  ontowfo 5128  1-1-ontowf1o 5129
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-br 3937  df-opab 3997  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137
This theorem is referenced by:  dif1en  6780
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