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Theorem elunirn 7271
Description: Membership in the union of the range of a function. See elunirnALT 7272 for a shorter proof which uses ax-pow 5371. See elfvunirn 6939 for a more general version of the reverse direction. (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 4915 . 2 (𝐴 ran 𝐹 ↔ ∃𝑦(𝐴𝑦𝑦 ∈ ran 𝐹))
2 funfn 6598 . . . . . . . 8 (Fun 𝐹𝐹 Fn dom 𝐹)
3 fvelrnb 6969 . . . . . . . 8 (𝐹 Fn dom 𝐹 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹(𝐹𝑥) = 𝑦))
42, 3sylbi 217 . . . . . . 7 (Fun 𝐹 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹(𝐹𝑥) = 𝑦))
54anbi2d 630 . . . . . 6 (Fun 𝐹 → ((𝐴𝑦𝑦 ∈ ran 𝐹) ↔ (𝐴𝑦 ∧ ∃𝑥 ∈ dom 𝐹(𝐹𝑥) = 𝑦)))
6 r19.42v 3189 . . . . . 6 (∃𝑥 ∈ dom 𝐹(𝐴𝑦 ∧ (𝐹𝑥) = 𝑦) ↔ (𝐴𝑦 ∧ ∃𝑥 ∈ dom 𝐹(𝐹𝑥) = 𝑦))
75, 6bitr4di 289 . . . . 5 (Fun 𝐹 → ((𝐴𝑦𝑦 ∈ ran 𝐹) ↔ ∃𝑥 ∈ dom 𝐹(𝐴𝑦 ∧ (𝐹𝑥) = 𝑦)))
8 eleq2 2828 . . . . . . 7 ((𝐹𝑥) = 𝑦 → (𝐴 ∈ (𝐹𝑥) ↔ 𝐴𝑦))
98biimparc 479 . . . . . 6 ((𝐴𝑦 ∧ (𝐹𝑥) = 𝑦) → 𝐴 ∈ (𝐹𝑥))
109reximi 3082 . . . . 5 (∃𝑥 ∈ dom 𝐹(𝐴𝑦 ∧ (𝐹𝑥) = 𝑦) → ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹𝑥))
117, 10biimtrdi 253 . . . 4 (Fun 𝐹 → ((𝐴𝑦𝑦 ∈ ran 𝐹) → ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹𝑥)))
1211exlimdv 1931 . . 3 (Fun 𝐹 → (∃𝑦(𝐴𝑦𝑦 ∈ ran 𝐹) → ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹𝑥)))
13 fvelrn 7096 . . . . . . 7 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (𝐹𝑥) ∈ ran 𝐹)
1413a1d 25 . . . . . 6 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (𝐴 ∈ (𝐹𝑥) → (𝐹𝑥) ∈ ran 𝐹))
1514ancld 550 . . . . 5 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (𝐴 ∈ (𝐹𝑥) → (𝐴 ∈ (𝐹𝑥) ∧ (𝐹𝑥) ∈ ran 𝐹)))
16 fvex 6920 . . . . . 6 (𝐹𝑥) ∈ V
17 eleq2 2828 . . . . . . 7 (𝑦 = (𝐹𝑥) → (𝐴𝑦𝐴 ∈ (𝐹𝑥)))
18 eleq1 2827 . . . . . . 7 (𝑦 = (𝐹𝑥) → (𝑦 ∈ ran 𝐹 ↔ (𝐹𝑥) ∈ ran 𝐹))
1917, 18anbi12d 632 . . . . . 6 (𝑦 = (𝐹𝑥) → ((𝐴𝑦𝑦 ∈ ran 𝐹) ↔ (𝐴 ∈ (𝐹𝑥) ∧ (𝐹𝑥) ∈ ran 𝐹)))
2016, 19spcev 3606 . . . . 5 ((𝐴 ∈ (𝐹𝑥) ∧ (𝐹𝑥) ∈ ran 𝐹) → ∃𝑦(𝐴𝑦𝑦 ∈ ran 𝐹))
2115, 20syl6 35 . . . 4 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (𝐴 ∈ (𝐹𝑥) → ∃𝑦(𝐴𝑦𝑦 ∈ ran 𝐹)))
2221rexlimdva 3153 . . 3 (Fun 𝐹 → (∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹𝑥) → ∃𝑦(𝐴𝑦𝑦 ∈ ran 𝐹)))
2312, 22impbid 212 . 2 (Fun 𝐹 → (∃𝑦(𝐴𝑦𝑦 ∈ ran 𝐹) ↔ ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹𝑥)))
241, 23bitrid 283 1 (Fun 𝐹 → (𝐴 ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹𝑥)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wex 1776  wcel 2106  wrex 3068   cuni 4912  dom cdm 5689  ran crn 5690  Fun wfun 6557   Fn wfn 6558  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571
This theorem is referenced by:  elunirn2OLD  7273  fnunirn  7274  fin23lem30  10380  ustn0  24245  ustbas  24252  elunirnmbfm  34233  fourierdlem70  46132  fourierdlem71  46133  fourierdlem80  46142
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