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Theorem chfnrn 6554
Description: The range of a choice function (a function that chooses an element from each member of its domain) is included in the union of its domain. (Contributed by NM, 31-Aug-1999.)
Assertion
Ref Expression
chfnrn ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝑥) → ran 𝐹 𝐴)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹

Proof of Theorem chfnrn
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 fvelrnb 6468 . . . . 5 (𝐹 Fn 𝐴 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥𝐴 (𝐹𝑥) = 𝑦))
21biimpd 221 . . . 4 (𝐹 Fn 𝐴 → (𝑦 ∈ ran 𝐹 → ∃𝑥𝐴 (𝐹𝑥) = 𝑦))
3 eleq1 2866 . . . . . . 7 ((𝐹𝑥) = 𝑦 → ((𝐹𝑥) ∈ 𝑥𝑦𝑥))
43biimpcd 241 . . . . . 6 ((𝐹𝑥) ∈ 𝑥 → ((𝐹𝑥) = 𝑦𝑦𝑥))
54ralimi 3133 . . . . 5 (∀𝑥𝐴 (𝐹𝑥) ∈ 𝑥 → ∀𝑥𝐴 ((𝐹𝑥) = 𝑦𝑦𝑥))
6 rexim 3188 . . . . 5 (∀𝑥𝐴 ((𝐹𝑥) = 𝑦𝑦𝑥) → (∃𝑥𝐴 (𝐹𝑥) = 𝑦 → ∃𝑥𝐴 𝑦𝑥))
75, 6syl 17 . . . 4 (∀𝑥𝐴 (𝐹𝑥) ∈ 𝑥 → (∃𝑥𝐴 (𝐹𝑥) = 𝑦 → ∃𝑥𝐴 𝑦𝑥))
82, 7sylan9 504 . . 3 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝑥) → (𝑦 ∈ ran 𝐹 → ∃𝑥𝐴 𝑦𝑥))
9 eluni2 4632 . . 3 (𝑦 𝐴 ↔ ∃𝑥𝐴 𝑦𝑥)
108, 9syl6ibr 244 . 2 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝑥) → (𝑦 ∈ ran 𝐹𝑦 𝐴))
1110ssrdv 3804 1 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝑥) → ran 𝐹 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  wral 3089  wrex 3090  wss 3769   cuni 4628  ran crn 5313   Fn wfn 6096  cfv 6101
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-iota 6064  df-fun 6103  df-fn 6104  df-fv 6109
This theorem is referenced by:  stoweidlem59  41015
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