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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 6153 | . . . . 5 ⊢ (𝐹 Fn 𝐴 → (𝑦 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦)) | |
2 | 1 | biimpd 218 | . . . 4 ⊢ (𝐹 Fn 𝐴 → (𝑦 ∈ ran 𝐹 → ∃𝑥 ∈ 𝐴 (𝐹‘𝑥) = 𝑦)) |
3 | eleq1 2676 | . . . . . . 7 ⊢ ((𝐹‘𝑥) = 𝑦 → ((𝐹‘𝑥) ∈ 𝑥 ↔ 𝑦 ∈ 𝑥)) | |
4 | 3 | biimpcd 238 | . . . . . 6 ⊢ ((𝐹‘𝑥) ∈ 𝑥 → ((𝐹‘𝑥) = 𝑦 → 𝑦 ∈ 𝑥)) |
5 | 4 | ralimi 2936 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝑥 → ∀𝑥 ∈ 𝐴 ((𝐹‘𝑥) = 𝑦 → 𝑦 ∈ 𝑥)) |
6 | rexim 2991 | . . . . 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 687 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝑥) → (𝑦 ∈ ran 𝐹 → ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥)) |
9 | eluni2 4376 | . . 3 ⊢ (𝑦 ∈ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝑥) | |
10 | 8, 9 | syl6ibr 241 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝑥) → (𝑦 ∈ ran 𝐹 → 𝑦 ∈ ∪ 𝐴)) |
11 | 10 | ssrdv 3574 | 1 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝑥) → ran 𝐹 ⊆ ∪ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 ⊆ wss 3540 ∪ cuni 4372 ran crn 5039 Fn wfn 5799 ‘cfv 5804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fn 5807 df-fv 5812 |
This theorem is referenced by: stoweidlem59 38952 |
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