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

Theorem unpreima 5217
Description: Preimage of a union. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
unpreima (Fun 𝐹 → (𝐹 “ (AB)) = ((𝐹A) ∪ (𝐹B)))

Proof of Theorem unpreima
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 funfn 4857 . 2 (Fun 𝐹𝐹 Fn dom 𝐹)
2 elpreima 5211 . . . 4 (𝐹 Fn dom 𝐹 → (x (𝐹 “ (AB)) ↔ (x dom 𝐹 (𝐹x) (AB))))
3 elun 3061 . . . . . 6 (x ((𝐹A) ∪ (𝐹B)) ↔ (x (𝐹A) x (𝐹B)))
4 elpreima 5211 . . . . . . 7 (𝐹 Fn dom 𝐹 → (x (𝐹A) ↔ (x dom 𝐹 (𝐹x) A)))
5 elpreima 5211 . . . . . . 7 (𝐹 Fn dom 𝐹 → (x (𝐹B) ↔ (x dom 𝐹 (𝐹x) B)))
64, 5orbi12d 694 . . . . . 6 (𝐹 Fn dom 𝐹 → ((x (𝐹A) x (𝐹B)) ↔ ((x dom 𝐹 (𝐹x) A) (x dom 𝐹 (𝐹x) B))))
73, 6syl5bb 181 . . . . 5 (𝐹 Fn dom 𝐹 → (x ((𝐹A) ∪ (𝐹B)) ↔ ((x dom 𝐹 (𝐹x) A) (x dom 𝐹 (𝐹x) B))))
8 elun 3061 . . . . . . 7 ((𝐹x) (AB) ↔ ((𝐹x) A (𝐹x) B))
98anbi2i 433 . . . . . 6 ((x dom 𝐹 (𝐹x) (AB)) ↔ (x dom 𝐹 ((𝐹x) A (𝐹x) B)))
10 andi 719 . . . . . 6 ((x dom 𝐹 ((𝐹x) A (𝐹x) B)) ↔ ((x dom 𝐹 (𝐹x) A) (x dom 𝐹 (𝐹x) B)))
119, 10bitri 173 . . . . 5 ((x dom 𝐹 (𝐹x) (AB)) ↔ ((x dom 𝐹 (𝐹x) A) (x dom 𝐹 (𝐹x) B)))
127, 11syl6rbbr 188 . . . 4 (𝐹 Fn dom 𝐹 → ((x dom 𝐹 (𝐹x) (AB)) ↔ x ((𝐹A) ∪ (𝐹B))))
132, 12bitrd 177 . . 3 (𝐹 Fn dom 𝐹 → (x (𝐹 “ (AB)) ↔ x ((𝐹A) ∪ (𝐹B))))
1413eqrdv 2020 . 2 (𝐹 Fn dom 𝐹 → (𝐹 “ (AB)) = ((𝐹A) ∪ (𝐹B)))
151, 14sylbi 114 1 (Fun 𝐹 → (𝐹 “ (AB)) = ((𝐹A) ∪ (𝐹B)))
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wo 616   = wceq 1228   wcel 1374  cun 2892  ccnv 4271  dom cdm 4272  cima 4275  Fun wfun 4823   Fn wfn 4824  cfv 4829
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-v 2537  df-sbc 2742  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-br 3739  df-opab 3793  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-iota 4794  df-fun 4831  df-fn 4832  df-fv 4837
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator