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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
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