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Theorem resasplit 5615
 Description: If two functions agree on their common domain, express their union as a union of three functions with pairwise disjoint domains. (Contributed by Stefan O'Rear, 9-Oct-2014.)
Assertion
Ref Expression
resasplit

Proof of Theorem resasplit
StepHypRef Expression
1 fnresdm 5556 . . . 4
2 fnresdm 5556 . . . 4
3 uneq12 3498 . . . 4
41, 2, 3syl2an 465 . . 3
6 simp3 960 . . . . . . 7
76uneq1d 3502 . . . . . 6
87uneq2d 3503 . . . . 5
9 inundif 3708 . . . . . . . 8
109reseq2i 5145 . . . . . . 7
11 resundi 5162 . . . . . . 7
1210, 11eqtr3i 2460 . . . . . 6
13 incom 3535 . . . . . . . . . 10
1413uneq1i 3499 . . . . . . . . 9
15 inundif 3708 . . . . . . . . 9
1614, 15eqtri 2458 . . . . . . . 8
1716reseq2i 5145 . . . . . . 7
18 resundi 5162 . . . . . . 7
1917, 18eqtr3i 2460 . . . . . 6
2012, 19uneq12i 3501 . . . . 5
218, 20syl6reqr 2489 . . . 4
22 un4 3509 . . . 4
2321, 22syl6eq 2486 . . 3
24 unidm 3492 . . . 4
2524uneq1i 3499 . . 3
2623, 25syl6eq 2486 . 2
275, 26eqtr3d 2472 1
 Colors of variables: wff set class Syntax hints:   wi 4   w3a 937   wceq 1653   cdif 3319   cun 3320   cin 3321   cres 4882   wfn 5451 This theorem is referenced by:  fresaun  5616  fresaunres2  5617 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4215  df-opab 4269  df-xp 4886  df-rel 4887  df-dm 4890  df-res 4892  df-fun 5458  df-fn 5459
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