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Theorem chfnrn 5305
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 5248 . . . . 5 (𝐹 Fn 𝐴 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥𝐴 (𝐹𝑥) = 𝑦))
21biimpd 136 . . . 4 (𝐹 Fn 𝐴 → (𝑦 ∈ ran 𝐹 → ∃𝑥𝐴 (𝐹𝑥) = 𝑦))
3 eleq1 2116 . . . . . . 7 ((𝐹𝑥) = 𝑦 → ((𝐹𝑥) ∈ 𝑥𝑦𝑥))
43biimpcd 152 . . . . . 6 ((𝐹𝑥) ∈ 𝑥 → ((𝐹𝑥) = 𝑦𝑦𝑥))
54ralimi 2401 . . . . 5 (∀𝑥𝐴 (𝐹𝑥) ∈ 𝑥 → ∀𝑥𝐴 ((𝐹𝑥) = 𝑦𝑦𝑥))
6 rexim 2430 . . . . 5 (∀𝑥𝐴 ((𝐹𝑥) = 𝑦𝑦𝑥) → (∃𝑥𝐴 (𝐹𝑥) = 𝑦 → ∃𝑥𝐴 𝑦𝑥))
75, 6syl 14 . . . 4 (∀𝑥𝐴 (𝐹𝑥) ∈ 𝑥 → (∃𝑥𝐴 (𝐹𝑥) = 𝑦 → ∃𝑥𝐴 𝑦𝑥))
82, 7sylan9 395 . . 3 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝑥) → (𝑦 ∈ ran 𝐹 → ∃𝑥𝐴 𝑦𝑥))
9 eluni2 3611 . . 3 (𝑦 𝐴 ↔ ∃𝑥𝐴 𝑦𝑥)
108, 9syl6ibr 155 . 2 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝑥) → (𝑦 ∈ ran 𝐹𝑦 𝐴))
1110ssrdv 2978 1 ((𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝑥) → ran 𝐹 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1259  wcel 1409  wral 2323  wrex 2324  wss 2944   cuni 3607  ran crn 4373   Fn wfn 4924  cfv 4929
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-sbc 2787  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-br 3792  df-opab 3846  df-mpt 3847  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-iota 4894  df-fun 4931  df-fn 4932  df-fv 4937
This theorem is referenced by: (None)
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