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

Theorem elunirn 5430
Description: Membership in the union of the range of a function. (Contributed by NM, 24-Sep-2006.)
Assertion
Ref Expression
elunirn (Fun 𝐹 → (𝐴 ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹𝑥)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹

Proof of Theorem elunirn
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 eluni 3608 . 2 (𝐴 ran 𝐹 ↔ ∃𝑦(𝐴𝑦𝑦 ∈ ran 𝐹))
2 funfn 4956 . . . . . . . 8 (Fun 𝐹𝐹 Fn dom 𝐹)
3 fvelrnb 5246 . . . . . . . 8 (𝐹 Fn dom 𝐹 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹(𝐹𝑥) = 𝑦))
42, 3sylbi 118 . . . . . . 7 (Fun 𝐹 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹(𝐹𝑥) = 𝑦))
54anbi2d 445 . . . . . 6 (Fun 𝐹 → ((𝐴𝑦𝑦 ∈ ran 𝐹) ↔ (𝐴𝑦 ∧ ∃𝑥 ∈ dom 𝐹(𝐹𝑥) = 𝑦)))
6 r19.42v 2482 . . . . . 6 (∃𝑥 ∈ dom 𝐹(𝐴𝑦 ∧ (𝐹𝑥) = 𝑦) ↔ (𝐴𝑦 ∧ ∃𝑥 ∈ dom 𝐹(𝐹𝑥) = 𝑦))
75, 6syl6bbr 191 . . . . 5 (Fun 𝐹 → ((𝐴𝑦𝑦 ∈ ran 𝐹) ↔ ∃𝑥 ∈ dom 𝐹(𝐴𝑦 ∧ (𝐹𝑥) = 𝑦)))
8 eleq2 2115 . . . . . . 7 ((𝐹𝑥) = 𝑦 → (𝐴 ∈ (𝐹𝑥) ↔ 𝐴𝑦))
98biimparc 287 . . . . . 6 ((𝐴𝑦 ∧ (𝐹𝑥) = 𝑦) → 𝐴 ∈ (𝐹𝑥))
109reximi 2431 . . . . 5 (∃𝑥 ∈ dom 𝐹(𝐴𝑦 ∧ (𝐹𝑥) = 𝑦) → ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹𝑥))
117, 10syl6bi 156 . . . 4 (Fun 𝐹 → ((𝐴𝑦𝑦 ∈ ran 𝐹) → ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹𝑥)))
1211exlimdv 1714 . . 3 (Fun 𝐹 → (∃𝑦(𝐴𝑦𝑦 ∈ ran 𝐹) → ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹𝑥)))
13 fvelrn 5323 . . . . 5 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (𝐹𝑥) ∈ ran 𝐹)
14 funfvex 5217 . . . . . 6 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (𝐹𝑥) ∈ V)
15 eleq2 2115 . . . . . . . 8 (𝑦 = (𝐹𝑥) → (𝐴𝑦𝐴 ∈ (𝐹𝑥)))
16 eleq1 2114 . . . . . . . 8 (𝑦 = (𝐹𝑥) → (𝑦 ∈ ran 𝐹 ↔ (𝐹𝑥) ∈ ran 𝐹))
1715, 16anbi12d 450 . . . . . . 7 (𝑦 = (𝐹𝑥) → ((𝐴𝑦𝑦 ∈ ran 𝐹) ↔ (𝐴 ∈ (𝐹𝑥) ∧ (𝐹𝑥) ∈ ran 𝐹)))
1817spcegv 2656 . . . . . 6 ((𝐹𝑥) ∈ V → ((𝐴 ∈ (𝐹𝑥) ∧ (𝐹𝑥) ∈ ran 𝐹) → ∃𝑦(𝐴𝑦𝑦 ∈ ran 𝐹)))
1914, 18syl 14 . . . . 5 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ((𝐴 ∈ (𝐹𝑥) ∧ (𝐹𝑥) ∈ ran 𝐹) → ∃𝑦(𝐴𝑦𝑦 ∈ ran 𝐹)))
2013, 19mpan2d 412 . . . 4 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (𝐴 ∈ (𝐹𝑥) → ∃𝑦(𝐴𝑦𝑦 ∈ ran 𝐹)))
2120rexlimdva 2448 . . 3 (Fun 𝐹 → (∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹𝑥) → ∃𝑦(𝐴𝑦𝑦 ∈ ran 𝐹)))
2212, 21impbid 124 . 2 (Fun 𝐹 → (∃𝑦(𝐴𝑦𝑦 ∈ ran 𝐹) ↔ ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹𝑥)))
231, 22syl5bb 185 1 (Fun 𝐹 → (𝐴 ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹𝑥)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1257  wex 1395  wcel 1407  wrex 2322  Vcvv 2572   cuni 3605  dom cdm 4370  ran crn 4371  Fun wfun 4921   Fn wfn 4922  cfv 4927
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-14 1419  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036  ax-sep 3900  ax-pow 3952  ax-pr 3969
This theorem depends on definitions:  df-bi 114  df-3an 896  df-tru 1260  df-nf 1364  df-sb 1660  df-eu 1917  df-mo 1918  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ral 2326  df-rex 2327  df-v 2574  df-sbc 2785  df-un 2947  df-in 2949  df-ss 2956  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3606  df-br 3790  df-opab 3844  df-mpt 3845  df-id 4055  df-xp 4376  df-rel 4377  df-cnv 4378  df-co 4379  df-dm 4380  df-rn 4381  df-iota 4892  df-fun 4929  df-fn 4930  df-fv 4935
This theorem is referenced by:  fnunirn  5431
  Copyright terms: Public domain W3C validator