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Theorem elunirn 5348
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 3574 . 2 (A ran 𝐹y(A y y ran 𝐹))
2 funfn 4874 . . . . . . . 8 (Fun 𝐹𝐹 Fn dom 𝐹)
3 fvelrnb 5164 . . . . . . . 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 5241 . . . . 5 ((Fun 𝐹 x dom 𝐹) → (𝐹x) ran 𝐹)
14 funfvex 5135 . . . . . 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 3571  dom cdm 4288  ran crn 4289  Fun wfun 4839   Fn wfn 4840  cfv 4845
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 3866  ax-pow 3918  ax-pr 3935
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 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853
This theorem is referenced by:  fnunirn  5349
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