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

Theorem imaundi 4679
Description: Distributive law for image over union. Theorem 35 of [Suppes] p. 65. (Contributed by NM, 30-Sep-2002.)
Assertion
Ref Expression
imaundi (A “ (B𝐶)) = ((AB) ∪ (A𝐶))

Proof of Theorem imaundi
StepHypRef Expression
1 resundi 4568 . . . 4 (A ↾ (B𝐶)) = ((AB) ∪ (A𝐶))
21rneqi 4505 . . 3 ran (A ↾ (B𝐶)) = ran ((AB) ∪ (A𝐶))
3 rnun 4675 . . 3 ran ((AB) ∪ (A𝐶)) = (ran (AB) ∪ ran (A𝐶))
42, 3eqtri 2057 . 2 ran (A ↾ (B𝐶)) = (ran (AB) ∪ ran (A𝐶))
5 df-ima 4301 . 2 (A “ (B𝐶)) = ran (A ↾ (B𝐶))
6 df-ima 4301 . . 3 (AB) = ran (AB)
7 df-ima 4301 . . 3 (A𝐶) = ran (A𝐶)
86, 7uneq12i 3089 . 2 ((AB) ∪ (A𝐶)) = (ran (AB) ∪ ran (A𝐶))
94, 5, 83eqtr4i 2067 1 (A “ (B𝐶)) = ((AB) ∪ (A𝐶))
Colors of variables: wff set class
Syntax hints:   = wceq 1242  cun 2909  ran crn 4289  cres 4290  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-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-op 3376  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:  fnimapr  5176
  Copyright terms: Public domain W3C validator