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Theorem chfnrn 7043
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 6945 . . . . 5 (𝐹 Fn 𝐴 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥𝐴 (𝐹𝑥) = 𝑦))
21biimpd 228 . . . 4 (𝐹 Fn 𝐴 → (𝑦 ∈ ran 𝐹 → ∃𝑥𝐴 (𝐹𝑥) = 𝑦))
3 eleq1 2815 . . . . . . 7 ((𝐹𝑥) = 𝑦 → ((𝐹𝑥) ∈ 𝑥𝑦𝑥))
43biimpcd 248 . . . . . 6 ((𝐹𝑥) ∈ 𝑥 → ((𝐹𝑥) = 𝑦𝑦𝑥))
54ralimi 3077 . . . . 5 (∀𝑥𝐴 (𝐹𝑥) ∈ 𝑥 → ∀𝑥𝐴 ((𝐹𝑥) = 𝑦𝑦𝑥))
6 rexim 3081 . . . . 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 4906 . . 3 (𝑦 𝐴 ↔ ∃𝑥𝐴 𝑦𝑥)
108, 9imbitrrdi 251 . 2 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝑥) → (𝑦 ∈ ran 𝐹𝑦 𝐴))
1110ssrdv 3983 1 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝑥) → ran 𝐹 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  wral 3055  wrex 3064  wss 3943   cuni 4902  ran crn 5670   Fn wfn 6531  cfv 6536
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-iota 6488  df-fun 6538  df-fn 6539  df-fv 6544
This theorem is referenced by:  stoweidlem59  45328
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