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Theorem elunirn 5405
 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 3583 . 2 (𝐴 ran 𝐹 ↔ ∃𝑦(𝐴𝑦𝑦 ∈ ran 𝐹))
2 funfn 4931 . . . . . . . 8 (Fun 𝐹𝐹 Fn dom 𝐹)
3 fvelrnb 5221 . . . . . . . 8 (𝐹 Fn dom 𝐹 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹(𝐹𝑥) = 𝑦))
42, 3sylbi 114 . . . . . . 7 (Fun 𝐹 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹(𝐹𝑥) = 𝑦))
54anbi2d 437 . . . . . 6 (Fun 𝐹 → ((𝐴𝑦𝑦 ∈ ran 𝐹) ↔ (𝐴𝑦 ∧ ∃𝑥 ∈ dom 𝐹(𝐹𝑥) = 𝑦)))
6 r19.42v 2467 . . . . . 6 (∃𝑥 ∈ dom 𝐹(𝐴𝑦 ∧ (𝐹𝑥) = 𝑦) ↔ (𝐴𝑦 ∧ ∃𝑥 ∈ dom 𝐹(𝐹𝑥) = 𝑦))
75, 6syl6bbr 187 . . . . 5 (Fun 𝐹 → ((𝐴𝑦𝑦 ∈ ran 𝐹) ↔ ∃𝑥 ∈ dom 𝐹(𝐴𝑦 ∧ (𝐹𝑥) = 𝑦)))
8 eleq2 2101 . . . . . . 7 ((𝐹𝑥) = 𝑦 → (𝐴 ∈ (𝐹𝑥) ↔ 𝐴𝑦))
98biimparc 283 . . . . . 6 ((𝐴𝑦 ∧ (𝐹𝑥) = 𝑦) → 𝐴 ∈ (𝐹𝑥))
109reximi 2416 . . . . 5 (∃𝑥 ∈ dom 𝐹(𝐴𝑦 ∧ (𝐹𝑥) = 𝑦) → ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹𝑥))
117, 10syl6bi 152 . . . 4 (Fun 𝐹 → ((𝐴𝑦𝑦 ∈ ran 𝐹) → ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹𝑥)))
1211exlimdv 1700 . . 3 (Fun 𝐹 → (∃𝑦(𝐴𝑦𝑦 ∈ ran 𝐹) → ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹𝑥)))
13 fvelrn 5298 . . . . 5 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (𝐹𝑥) ∈ ran 𝐹)
14 funfvex 5192 . . . . . 6 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (𝐹𝑥) ∈ V)
15 eleq2 2101 . . . . . . . 8 (𝑦 = (𝐹𝑥) → (𝐴𝑦𝐴 ∈ (𝐹𝑥)))
16 eleq1 2100 . . . . . . . 8 (𝑦 = (𝐹𝑥) → (𝑦 ∈ ran 𝐹 ↔ (𝐹𝑥) ∈ ran 𝐹))
1715, 16anbi12d 442 . . . . . . 7 (𝑦 = (𝐹𝑥) → ((𝐴𝑦𝑦 ∈ ran 𝐹) ↔ (𝐴 ∈ (𝐹𝑥) ∧ (𝐹𝑥) ∈ ran 𝐹)))
1817spcegv 2641 . . . . . 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 404 . . . 4 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (𝐴 ∈ (𝐹𝑥) → ∃𝑦(𝐴𝑦𝑦 ∈ ran 𝐹)))
2120rexlimdva 2433 . . 3 (Fun 𝐹 → (∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹𝑥) → ∃𝑦(𝐴𝑦𝑦 ∈ ran 𝐹)))
2212, 21impbid 120 . 2 (Fun 𝐹 → (∃𝑦(𝐴𝑦𝑦 ∈ ran 𝐹) ↔ ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹𝑥)))
231, 22syl5bb 181 1 (Fun 𝐹 → (𝐴 ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹𝑥)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1243  ∃wex 1381   ∈ wcel 1393  ∃wrex 2307  Vcvv 2557  ∪ cuni 3580  dom cdm 4345  ran crn 4346  Fun wfun 4896   Fn wfn 4897  ‘cfv 4902 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-iota 4867  df-fun 4904  df-fn 4905  df-fv 4910 This theorem is referenced by:  fnunirn  5406
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