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Theorem resasplit 5765
 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 5695 . . . 4
2 fnresdm 5695 . . . 4
3 uneq12 3574 . . . 4
41, 2, 3syl2an 485 . . 3
543adant3 1050 . 2
6 simp3 1032 . . . . . . 7
76uneq1d 3578 . . . . . 6
87uneq2d 3579 . . . . 5
9 inundif 3836 . . . . . . . 8
109reseq2i 5108 . . . . . . 7
11 resundi 5124 . . . . . . 7
1210, 11eqtr3i 2495 . . . . . 6
13 incom 3616 . . . . . . . . . 10
1413uneq1i 3575 . . . . . . . . 9
15 inundif 3836 . . . . . . . . 9
1614, 15eqtri 2493 . . . . . . . 8
1716reseq2i 5108 . . . . . . 7
18 resundi 5124 . . . . . . 7
1917, 18eqtr3i 2495 . . . . . 6
2012, 19uneq12i 3577 . . . . 5
218, 20syl6reqr 2524 . . . 4
22 un4 3585 . . . 4
2321, 22syl6eq 2521 . . 3
24 unidm 3568 . . . 4
2524uneq1i 3575 . . 3
2623, 25syl6eq 2521 . 2
275, 26eqtr3d 2507 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   w3a 1007   wceq 1452   cdif 3387   cun 3388   cin 3389   cres 4841   wfn 5584 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-br 4396  df-opab 4455  df-xp 4845  df-rel 4846  df-dm 4849  df-res 4851  df-fun 5591  df-fn 5592 This theorem is referenced by:  fresaun  5766  fresaunres2  5767
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