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Theorem elunirn 6413
 Description: Membership in the union of the range of a function. See elunirnALT 6414 for a shorter proof which uses ax-pow 4769. (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 4375 . 2 (𝐴 ran 𝐹 ↔ ∃𝑦(𝐴𝑦𝑦 ∈ ran 𝐹))
2 funfn 5833 . . . . . . . 8 (Fun 𝐹𝐹 Fn dom 𝐹)
3 fvelrnb 6153 . . . . . . . 8 (𝐹 Fn dom 𝐹 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹(𝐹𝑥) = 𝑦))
42, 3sylbi 206 . . . . . . 7 (Fun 𝐹 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹(𝐹𝑥) = 𝑦))
54anbi2d 736 . . . . . 6 (Fun 𝐹 → ((𝐴𝑦𝑦 ∈ ran 𝐹) ↔ (𝐴𝑦 ∧ ∃𝑥 ∈ dom 𝐹(𝐹𝑥) = 𝑦)))
6 r19.42v 3073 . . . . . 6 (∃𝑥 ∈ dom 𝐹(𝐴𝑦 ∧ (𝐹𝑥) = 𝑦) ↔ (𝐴𝑦 ∧ ∃𝑥 ∈ dom 𝐹(𝐹𝑥) = 𝑦))
75, 6syl6bbr 277 . . . . 5 (Fun 𝐹 → ((𝐴𝑦𝑦 ∈ ran 𝐹) ↔ ∃𝑥 ∈ dom 𝐹(𝐴𝑦 ∧ (𝐹𝑥) = 𝑦)))
8 eleq2 2677 . . . . . . 7 ((𝐹𝑥) = 𝑦 → (𝐴 ∈ (𝐹𝑥) ↔ 𝐴𝑦))
98biimparc 503 . . . . . 6 ((𝐴𝑦 ∧ (𝐹𝑥) = 𝑦) → 𝐴 ∈ (𝐹𝑥))
109reximi 2994 . . . . 5 (∃𝑥 ∈ dom 𝐹(𝐴𝑦 ∧ (𝐹𝑥) = 𝑦) → ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹𝑥))
117, 10syl6bi 242 . . . 4 (Fun 𝐹 → ((𝐴𝑦𝑦 ∈ ran 𝐹) → ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹𝑥)))
1211exlimdv 1848 . . 3 (Fun 𝐹 → (∃𝑦(𝐴𝑦𝑦 ∈ ran 𝐹) → ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹𝑥)))
13 fvelrn 6260 . . . . . . 7 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (𝐹𝑥) ∈ ran 𝐹)
1413a1d 25 . . . . . 6 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (𝐴 ∈ (𝐹𝑥) → (𝐹𝑥) ∈ ran 𝐹))
1514ancld 574 . . . . 5 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (𝐴 ∈ (𝐹𝑥) → (𝐴 ∈ (𝐹𝑥) ∧ (𝐹𝑥) ∈ ran 𝐹)))
16 fvex 6113 . . . . . 6 (𝐹𝑥) ∈ V
17 eleq2 2677 . . . . . . 7 (𝑦 = (𝐹𝑥) → (𝐴𝑦𝐴 ∈ (𝐹𝑥)))
18 eleq1 2676 . . . . . . 7 (𝑦 = (𝐹𝑥) → (𝑦 ∈ ran 𝐹 ↔ (𝐹𝑥) ∈ ran 𝐹))
1917, 18anbi12d 743 . . . . . 6 (𝑦 = (𝐹𝑥) → ((𝐴𝑦𝑦 ∈ ran 𝐹) ↔ (𝐴 ∈ (𝐹𝑥) ∧ (𝐹𝑥) ∈ ran 𝐹)))
2016, 19spcev 3273 . . . . 5 ((𝐴 ∈ (𝐹𝑥) ∧ (𝐹𝑥) ∈ ran 𝐹) → ∃𝑦(𝐴𝑦𝑦 ∈ ran 𝐹))
2115, 20syl6 34 . . . 4 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (𝐴 ∈ (𝐹𝑥) → ∃𝑦(𝐴𝑦𝑦 ∈ ran 𝐹)))
2221rexlimdva 3013 . . 3 (Fun 𝐹 → (∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹𝑥) → ∃𝑦(𝐴𝑦𝑦 ∈ ran 𝐹)))
2312, 22impbid 201 . 2 (Fun 𝐹 → (∃𝑦(𝐴𝑦𝑦 ∈ ran 𝐹) ↔ ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹𝑥)))
241, 23syl5bb 271 1 (Fun 𝐹 → (𝐴 ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹𝑥)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∃wrex 2897  ∪ cuni 4372  dom cdm 5038  ran crn 5039  Fun wfun 5798   Fn wfn 5799  ‘cfv 5804 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812 This theorem is referenced by:  fnunirn  6415  fin23lem30  9047  ustn0  21834  elrnust  21838  ustbas  21841  metuval  22164  elunirn2  28831  metidval  29261  pstmval  29266  elunirnmbfm  29642  fourierdlem70  39069  fourierdlem71  39070  fourierdlem80  39079
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