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Theorem elunirn 5346
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 3573 . 2 (A ran 𝐹y(A y y ran 𝐹))
2 funfn 4872 . . . . . . . 8 (Fun 𝐹𝐹 Fn dom 𝐹)
3 fvelrnb 5162 . . . . . . . 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 437 . . . . . 6 (Fun 𝐹 → ((A y y ran 𝐹) ↔ (A y x dom 𝐹(𝐹x) = y)))
6 r19.42v 2461 . . . . . 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 2098 . . . . . . 7 ((𝐹x) = y → (A (𝐹x) ↔ A y))
98biimparc 283 . . . . . 6 ((A y (𝐹x) = y) → A (𝐹x))
109reximi 2410 . . . . 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 1697 . . 3 (Fun 𝐹 → (y(A y y ran 𝐹) → x dom 𝐹 A (𝐹x)))
13 fvelrn 5239 . . . . 5 ((Fun 𝐹 x dom 𝐹) → (𝐹x) ran 𝐹)
14 funfvex 5133 . . . . . 6 ((Fun 𝐹 x dom 𝐹) → (𝐹x) V)
15 eleq2 2098 . . . . . . . 8 (y = (𝐹x) → (A yA (𝐹x)))
16 eleq1 2097 . . . . . . . 8 (y = (𝐹x) → (y ran 𝐹 ↔ (𝐹x) ran 𝐹))
1715, 16anbi12d 442 . . . . . . 7 (y = (𝐹x) → ((A y y ran 𝐹) ↔ (A (𝐹x) (𝐹x) ran 𝐹)))
1817spcegv 2635 . . . . . 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 404 . . . 4 ((Fun 𝐹 x dom 𝐹) → (A (𝐹x) → y(A y y ran 𝐹)))
2120rexlimdva 2427 . . 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 1242  wex 1378   wcel 1390  wrex 2301  Vcvv 2551   cuni 3570  dom cdm 4287  ran crn 4288  Fun wfun 4838   Fn wfn 4839  cfv 4844
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-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3865  ax-pow 3917  ax-pr 3934
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-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3352  df-sn 3372  df-pr 3373  df-op 3375  df-uni 3571  df-br 3755  df-opab 3809  df-mpt 3810  df-id 4020  df-xp 4293  df-rel 4294  df-cnv 4295  df-co 4296  df-dm 4297  df-rn 4298  df-iota 4809  df-fun 4846  df-fn 4847  df-fv 4852
This theorem is referenced by:  fnunirn  5347
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