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Mirrors > Home > MPE Home > Th. List > chfnrn | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
chfnrn | ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝑥) → ran 𝐹 ⊆ ∪ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvelrnb 6964 | . . . . 5 ⊢ (𝐹 Fn 𝐴 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦)) | |
2 | 1 | biimpd 228 | . . . 4 ⊢ (𝐹 Fn 𝐴 → (𝑦 ∈ ran 𝐹 → ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦)) |
3 | eleq1 2817 | . . . . . . 7 ⊢ ((𝐹‘𝑥) = 𝑦 → ((𝐹‘𝑥) ∈ 𝑥 ↔ 𝑦 ∈ 𝑥)) | |
4 | 3 | biimpcd 248 | . . . . . 6 ⊢ ((𝐹‘𝑥) ∈ 𝑥 → ((𝐹‘𝑥) = 𝑦 → 𝑦 ∈ 𝑥)) |
5 | 4 | ralimi 3080 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝑥 → ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 𝑦 → 𝑦 ∈ 𝑥)) |
6 | rexim 3084 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 𝑦 → 𝑦 ∈ 𝑥) → (∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦 → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥)) | |
7 | 5, 6 | syl 17 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝑥 → (∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦 → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥)) |
8 | 2, 7 | sylan9 506 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝑥) → (𝑦 ∈ ran 𝐹 → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥)) |
9 | eluni2 4916 | . . 3 ⊢ (𝑦 ∈ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥) | |
10 | 8, 9 | imbitrrdi 251 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝑥) → (𝑦 ∈ ran 𝐹 → 𝑦 ∈ ∪ 𝐴)) |
11 | 10 | ssrdv 3988 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝑥) → ran 𝐹 ⊆ ∪ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∀wral 3058 ∃wrex 3067 ⊆ wss 3949 ∪ cuni 4912 ran crn 5683 Fn wfn 6548 ‘cfv 6553 |
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 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-iota 6505 df-fun 6555 df-fn 6556 df-fv 6561 |
This theorem is referenced by: stoweidlem59 45476 |
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