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Theorem chfnrn 7050
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 6952 . . . . 5 (𝐹 Fn 𝐴 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥𝐴 (𝐹𝑥) = 𝑦))
21biimpd 228 . . . 4 (𝐹 Fn 𝐴 → (𝑦 ∈ ran 𝐹 → ∃𝑥𝐴 (𝐹𝑥) = 𝑦))
3 eleq1 2820 . . . . . . 7 ((𝐹𝑥) = 𝑦 → ((𝐹𝑥) ∈ 𝑥𝑦𝑥))
43biimpcd 248 . . . . . 6 ((𝐹𝑥) ∈ 𝑥 → ((𝐹𝑥) = 𝑦𝑦𝑥))
54ralimi 3082 . . . . 5 (∀𝑥𝐴 (𝐹𝑥) ∈ 𝑥 → ∀𝑥𝐴 ((𝐹𝑥) = 𝑦𝑦𝑥))
6 rexim 3086 . . . . 5 (∀𝑥𝐴 ((𝐹𝑥) = 𝑦𝑦𝑥) → (∃𝑥𝐴 (𝐹𝑥) = 𝑦 → ∃𝑥𝐴 𝑦𝑥))
75, 6syl 17 . . . 4 (∀𝑥𝐴 (𝐹𝑥) ∈ 𝑥 → (∃𝑥𝐴 (𝐹𝑥) = 𝑦 → ∃𝑥𝐴 𝑦𝑥))
82, 7sylan9 507 . . 3 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝑥) → (𝑦 ∈ ran 𝐹 → ∃𝑥𝐴 𝑦𝑥))
9 eluni2 4912 . . 3 (𝑦 𝐴 ↔ ∃𝑥𝐴 𝑦𝑥)
108, 9imbitrrdi 251 . 2 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝑥) → (𝑦 ∈ ran 𝐹𝑦 𝐴))
1110ssrdv 3988 1 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝑥) → ran 𝐹 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2105  wral 3060  wrex 3069  wss 3948   cuni 4908  ran crn 5677   Fn wfn 6538  cfv 6543
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-iota 6495  df-fun 6545  df-fn 6546  df-fv 6551
This theorem is referenced by:  stoweidlem59  45234
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