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Theorem fresaunres1 5741
 Description: From the union of two functions that agree on the domain overlap, either component can be recovered by restriction. (Contributed by Mario Carneiro, 16-Feb-2015.)
Assertion
Ref Expression
fresaunres1

Proof of Theorem fresaunres1
StepHypRef Expression
1 uncom 3587 . . 3
21reseq1i 5090 . 2
3 incom 3632 . . . . . 6
43reseq2i 5091 . . . . 5
53reseq2i 5091 . . . . 5
64, 5eqeq12i 2422 . . . 4
7 eqcom 2411 . . . 4
86, 7bitri 249 . . 3
9 fresaunres2 5740 . . . 4
1093com12 1201 . . 3
118, 10syl3an3b 1268 . 2
122, 11syl5eq 2455 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   w3a 974   wceq 1405   cun 3412   cin 3413   cres 4825  wf 5565 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pr 4630 This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-br 4396  df-opab 4454  df-xp 4829  df-rel 4830  df-dm 4833  df-res 4835  df-fun 5571  df-fn 5572  df-f 5573 This theorem is referenced by:  mapunen  7724  hashf1lem1  12553  ptuncnv  20600  resf1o  28000  cvmliftlem10  29591  aacllem  38860
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