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

Theorem elunirn 5326
 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 𝐹x 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 3553 . 2 (A ran 𝐹y(A y y ran 𝐹))
2 funfn 4853 . . . . . . . 8 (Fun 𝐹𝐹 Fn dom 𝐹)
3 fvelrnb 5142 . . . . . . . 8 (𝐹 Fn dom 𝐹 → (y ran 𝐹x dom 𝐹(𝐹x) = y))
42, 3sylbi 114 . . . . . . 7 (Fun 𝐹 → (y ran 𝐹x dom 𝐹(𝐹x) = y))
54anbi2d 440 . . . . . 6 (Fun 𝐹 → ((A y y ran 𝐹) ↔ (A y x dom 𝐹(𝐹x) = y)))
6 r19.42v 2441 . . . . . 6 (x dom 𝐹(A y (𝐹x) = y) ↔ (A y x dom 𝐹(𝐹x) = y))
75, 6syl6bbr 187 . . . . 5 (Fun 𝐹 → ((A y y ran 𝐹) ↔ x dom 𝐹(A y (𝐹x) = y)))
8 eleq2 2079 . . . . . . 7 ((𝐹x) = y → (A (𝐹x) ↔ A y))
98biimparc 283 . . . . . 6 ((A y (𝐹x) = y) → A (𝐹x))
109reximi 2390 . . . . 5 (x dom 𝐹(A y (𝐹x) = y) → x dom 𝐹 A (𝐹x))
117, 10syl6bi 152 . . . 4 (Fun 𝐹 → ((A y y ran 𝐹) → x dom 𝐹 A (𝐹x)))
1211exlimdv 1678 . . 3 (Fun 𝐹 → (y(A y y ran 𝐹) → x dom 𝐹 A (𝐹x)))
13 fvelrn 5219 . . . . 5 ((Fun 𝐹 x dom 𝐹) → (𝐹x) ran 𝐹)
14 funfvex 5113 . . . . . 6 ((Fun 𝐹 x dom 𝐹) → (𝐹x) V)
15 eleq2 2079 . . . . . . . 8 (y = (𝐹x) → (A yA (𝐹x)))
16 eleq1 2078 . . . . . . . 8 (y = (𝐹x) → (y ran 𝐹 ↔ (𝐹x) ran 𝐹))
1715, 16anbi12d 445 . . . . . . 7 (y = (𝐹x) → ((A y y ran 𝐹) ↔ (A (𝐹x) (𝐹x) ran 𝐹)))
1817spcegv 2614 . . . . . 6 ((𝐹x) V → ((A (𝐹x) (𝐹x) ran 𝐹) → y(A y y ran 𝐹)))
1914, 18syl 14 . . . . 5 ((Fun 𝐹 x dom 𝐹) → ((A (𝐹x) (𝐹x) ran 𝐹) → y(A y y ran 𝐹)))
2013, 19mpan2d 406 . . . 4 ((Fun 𝐹 x dom 𝐹) → (A (𝐹x) → y(A y y ran 𝐹)))
2120rexlimdva 2407 . . 3 (Fun 𝐹 → (x dom 𝐹 A (𝐹x) → y(A y y ran 𝐹)))
2212, 21impbid 120 . 2 (Fun 𝐹 → (y(A y y ran 𝐹) ↔ x dom 𝐹 A (𝐹x)))
231, 22syl5bb 181 1 (Fun 𝐹 → (A ran 𝐹x dom 𝐹 A (𝐹x)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1226  ∃wex 1358   ∈ wcel 1370  ∃wrex 2281  Vcvv 2531  ∪ cuni 3550  dom cdm 4268  ran crn 4269  Fun wfun 4819   Fn wfn 4820  ‘cfv 4825 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 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914 This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-sbc 2738  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-br 3735  df-opab 3789  df-mpt 3790  df-id 4000  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-rn 4279  df-iota 4790  df-fun 4827  df-fn 4828  df-fv 4833 This theorem is referenced by:  fnunirn  5327
 Copyright terms: Public domain W3C validator