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Theorem elunirn 7255
Description: Membership in the union of the range of a function. See elunirnALT 7256 for a shorter proof which uses ax-pow 5359. See elfvunirn 6923 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 4906 . 2 (𝐴 ran 𝐹 ↔ ∃𝑦(𝐴𝑦𝑦 ∈ ran 𝐹))
2 funfn 6577 . . . . . . . 8 (Fun 𝐹𝐹 Fn dom 𝐹)
3 fvelrnb 6953 . . . . . . . 8 (𝐹 Fn dom 𝐹 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹(𝐹𝑥) = 𝑦))
42, 3sylbi 216 . . . . . . 7 (Fun 𝐹 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹(𝐹𝑥) = 𝑦))
54anbi2d 628 . . . . . 6 (Fun 𝐹 → ((𝐴𝑦𝑦 ∈ ran 𝐹) ↔ (𝐴𝑦 ∧ ∃𝑥 ∈ dom 𝐹(𝐹𝑥) = 𝑦)))
6 r19.42v 3185 . . . . . 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 2817 . . . . . . 7 ((𝐹𝑥) = 𝑦 → (𝐴 ∈ (𝐹𝑥) ↔ 𝐴𝑦))
98biimparc 479 . . . . . 6 ((𝐴𝑦 ∧ (𝐹𝑥) = 𝑦) → 𝐴 ∈ (𝐹𝑥))
109reximi 3079 . . . . 5 (∃𝑥 ∈ dom 𝐹(𝐴𝑦 ∧ (𝐹𝑥) = 𝑦) → ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹𝑥))
117, 10biimtrdi 252 . . . 4 (Fun 𝐹 → ((𝐴𝑦𝑦 ∈ ran 𝐹) → ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹𝑥)))
1211exlimdv 1929 . . 3 (Fun 𝐹 → (∃𝑦(𝐴𝑦𝑦 ∈ ran 𝐹) → ∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹𝑥)))
13 fvelrn 7080 . . . . . . 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 6904 . . . . . 6 (𝐹𝑥) ∈ V
17 eleq2 2817 . . . . . . 7 (𝑦 = (𝐹𝑥) → (𝐴𝑦𝐴 ∈ (𝐹𝑥)))
18 eleq1 2816 . . . . . . 7 (𝑦 = (𝐹𝑥) → (𝑦 ∈ ran 𝐹 ↔ (𝐹𝑥) ∈ ran 𝐹))
1917, 18anbi12d 630 . . . . . 6 (𝑦 = (𝐹𝑥) → ((𝐴𝑦𝑦 ∈ ran 𝐹) ↔ (𝐴 ∈ (𝐹𝑥) ∧ (𝐹𝑥) ∈ ran 𝐹)))
2016, 19spcev 3591 . . . . 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 3150 . . 3 (Fun 𝐹 → (∃𝑥 ∈ dom 𝐹 𝐴 ∈ (𝐹𝑥) → ∃𝑦(𝐴𝑦𝑦 ∈ ran 𝐹)))
2312, 22impbid 211 . 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 205  wa 395   = wceq 1534  wex 1774  wcel 2099  wrex 3065   cuni 4903  dom cdm 5672  ran crn 5673  Fun wfun 6536   Fn wfn 6537  cfv 6542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-iota 6494  df-fun 6544  df-fn 6545  df-fv 6550
This theorem is referenced by:  elunirn2OLD  7257  fnunirn  7258  fin23lem30  10351  ustn0  24099  ustbas  24106  elunirnmbfm  33794  fourierdlem70  45477  fourierdlem71  45478  fourierdlem80  45487
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