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Theorem fun 5006
Description: The union of two functions with disjoint domains. (Contributed by NM, 22-Sep-2004.)
Assertion
Ref Expression
fun  F : --> C  G : --> D  i^i  (/)  F  u.  G :  u.  --> C  u.  D

Proof of Theorem fun
StepHypRef Expression
1 fnun 4948 . . . . 5  F  Fn  G  Fn  i^i  (/)  F  u.  G  Fn  u.
21expcom 109 . . . 4  i^i  (/)  F  Fn  G  Fn  F  u.  G  Fn  u.
3 rnun 4675 . . . . . 6  ran  F  u.  G  ran  F  u.  ran  G
4 unss12 3109 . . . . . 6  ran  F  C_  C  ran  G  C_  D  ran  F  u.  ran  G  C_  C  u.  D
53, 4syl5eqss 2983 . . . . 5  ran  F  C_  C  ran  G  C_  D  ran  F  u.  G  C_  C  u.  D
65a1i 9 . . . 4  i^i  (/)  ran  F  C_  C  ran  G  C_  D  ran  F  u.  G  C_  C  u.  D
72, 6anim12d 318 . . 3  i^i  (/)  F  Fn  G  Fn  ran  F  C_  C  ran  G  C_  D  F  u.  G  Fn  u.  ran  F  u.  G  C_  C  u.  D
8 df-f 4849 . . . . 5  F : --> C  F  Fn  ran  F 
C_  C
9 df-f 4849 . . . . 5  G : --> D  G  Fn  ran  G 
C_  D
108, 9anbi12i 433 . . . 4  F : --> C  G : --> D  F  Fn  ran  F  C_  C  G  Fn  ran  G  C_  D
11 an4 520 . . . 4  F  Fn  ran  F  C_  C  G  Fn  ran  G  C_  D  F  Fn  G  Fn  ran  F  C_  C  ran  G  C_  D
1210, 11bitri 173 . . 3  F : --> C  G : --> D  F  Fn  G  Fn  ran  F  C_  C  ran  G  C_  D
13 df-f 4849 . . 3  F  u.  G :  u.  --> C  u.  D  F  u.  G  Fn  u.  ran  F  u.  G  C_  C  u.  D
147, 12, 133imtr4g 194 . 2  i^i  (/)  F : --> C  G : --> D  F  u.  G :  u.  --> C  u.  D
1514impcom 116 1  F : --> C  G : --> D  i^i  (/)  F  u.  G :  u.  --> C  u.  D
Colors of variables: wff set class
Syntax hints:   wi 4   wa 97   wceq 1242    u. cun 2909    i^i cin 2910    C_ wss 2911   (/)c0 3218   ran crn 4289    Fn wfn 4840   -->wf 4841
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-fun 4847  df-fn 4848  df-f 4849
This theorem is referenced by:  fun2  5007  ftpg  5290  fsnunf  5305
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