ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  imaiun Structured version   Unicode version

Theorem imaiun 5342
Description: The image of an indexed union is the indexed union of the images. (Contributed by Mario Carneiro, 18-Jun-2014.)
Assertion
Ref Expression
imaiun 
" U_  C  U_  " C
Distinct variable group:   ,
Allowed substitution hints:   ()    C()

Proof of Theorem imaiun
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 rexcom4 2571 . . . 4  C  <. , 
>.  C  <. , 
>.
2 vex 2554 . . . . . 6 
_V
32elima3 4618 . . . . 5  " C  C  <. , 
>.
43rexbii 2325 . . . 4  " C  C  <. ,  >.
5 eliun 3652 . . . . . . 7  U_  C  C
65anbi1i 431 . . . . . 6  U_  C  <. ,  >.  C  <. , 
>.
7 r19.41v 2460 . . . . . 6  C  <. ,  >.  C  <. ,  >.
86, 7bitr4i 176 . . . . 5  U_  C  <. ,  >.  C  <. , 
>.
98exbii 1493 . . . 4  U_  C  <. , 
>.  C  <. , 
>.
101, 4, 93bitr4ri 202 . . 3  U_  C  <. , 
>.  " C
112elima3 4618 . . 3  " U_  C  U_  C  <. ,  >.
12 eliun 3652 . . 3  U_  " C  " C
1310, 11, 123bitr4i 201 . 2  " U_  C  U_  " C
1413eqriv 2034 1 
" U_  C  U_  " C
Colors of variables: wff set class
Syntax hints:   wa 97   wceq 1242  wex 1378   wcel 1390  wrex 2301   <.cop 3370   U_ciun 3648   "cima 4291
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-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-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-iun 3650  df-br 3756  df-opab 3810  df-xp 4294  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301
This theorem is referenced by:  imauni  5343  uniqs  6100
  Copyright terms: Public domain W3C validator