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Theorem chfnrn 6368
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 6282 . . . . 5 (𝐹 Fn 𝐴 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥𝐴 (𝐹𝑥) = 𝑦))
21biimpd 219 . . . 4 (𝐹 Fn 𝐴 → (𝑦 ∈ ran 𝐹 → ∃𝑥𝐴 (𝐹𝑥) = 𝑦))
3 eleq1 2718 . . . . . . 7 ((𝐹𝑥) = 𝑦 → ((𝐹𝑥) ∈ 𝑥𝑦𝑥))
43biimpcd 239 . . . . . 6 ((𝐹𝑥) ∈ 𝑥 → ((𝐹𝑥) = 𝑦𝑦𝑥))
54ralimi 2981 . . . . 5 (∀𝑥𝐴 (𝐹𝑥) ∈ 𝑥 → ∀𝑥𝐴 ((𝐹𝑥) = 𝑦𝑦𝑥))
6 rexim 3037 . . . . 5 (∀𝑥𝐴 ((𝐹𝑥) = 𝑦𝑦𝑥) → (∃𝑥𝐴 (𝐹𝑥) = 𝑦 → ∃𝑥𝐴 𝑦𝑥))
75, 6syl 17 . . . 4 (∀𝑥𝐴 (𝐹𝑥) ∈ 𝑥 → (∃𝑥𝐴 (𝐹𝑥) = 𝑦 → ∃𝑥𝐴 𝑦𝑥))
82, 7sylan9 690 . . 3 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝑥) → (𝑦 ∈ ran 𝐹 → ∃𝑥𝐴 𝑦𝑥))
9 eluni2 4472 . . 3 (𝑦 𝐴 ↔ ∃𝑥𝐴 𝑦𝑥)
108, 9syl6ibr 242 . 2 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝑥) → (𝑦 ∈ ran 𝐹𝑦 𝐴))
1110ssrdv 3642 1 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝑥) → ran 𝐹 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wcel 2030  wral 2941  wrex 2942  wss 3607   cuni 4468  ran crn 5144   Fn wfn 5921  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934
This theorem is referenced by:  stoweidlem59  40594
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