Mathbox for Alexander van der Vekens |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ffnafv | Structured version Visualization version GIF version |
Description: A function maps to a class to which all values belong, analogous to ffnfv 6295. (Contributed by Alexander van der Vekens, 25-May-2017.) |
Ref | Expression |
---|---|
ffnafv | ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹'''𝑥) ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ffn 5958 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → 𝐹 Fn 𝐴) | |
2 | fafvelrn 39899 | . . . 4 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝑥 ∈ 𝐴) → (𝐹'''𝑥) ∈ 𝐵) | |
3 | 2 | ralrimiva 2949 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 → ∀𝑥 ∈ 𝐴 (𝐹'''𝑥) ∈ 𝐵) |
4 | 1, 3 | jca 553 | . 2 ⊢ (𝐹:𝐴⟶𝐵 → (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹'''𝑥) ∈ 𝐵)) |
5 | simpl 472 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹'''𝑥) ∈ 𝐵) → 𝐹 Fn 𝐴) | |
6 | afvelrnb0 39893 | . . . . 5 ⊢ (𝐹 Fn 𝐴 → (𝑦 ∈ ran 𝐹 → ∃𝑥 ∈ 𝐴 (𝐹'''𝑥) = 𝑦)) | |
7 | nfra1 2925 | . . . . . 6 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 (𝐹'''𝑥) ∈ 𝐵 | |
8 | nfv 1830 | . . . . . 6 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐵 | |
9 | rsp 2913 | . . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 (𝐹'''𝑥) ∈ 𝐵 → (𝑥 ∈ 𝐴 → (𝐹'''𝑥) ∈ 𝐵)) | |
10 | eleq1 2676 | . . . . . . . 8 ⊢ ((𝐹'''𝑥) = 𝑦 → ((𝐹'''𝑥) ∈ 𝐵 ↔ 𝑦 ∈ 𝐵)) | |
11 | 10 | biimpcd 238 | . . . . . . 7 ⊢ ((𝐹'''𝑥) ∈ 𝐵 → ((𝐹'''𝑥) = 𝑦 → 𝑦 ∈ 𝐵)) |
12 | 9, 11 | syl6 34 | . . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 (𝐹'''𝑥) ∈ 𝐵 → (𝑥 ∈ 𝐴 → ((𝐹'''𝑥) = 𝑦 → 𝑦 ∈ 𝐵))) |
13 | 7, 8, 12 | rexlimd 3008 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (𝐹'''𝑥) ∈ 𝐵 → (∃𝑥 ∈ 𝐴 (𝐹'''𝑥) = 𝑦 → 𝑦 ∈ 𝐵)) |
14 | 6, 13 | sylan9 687 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹'''𝑥) ∈ 𝐵) → (𝑦 ∈ ran 𝐹 → 𝑦 ∈ 𝐵)) |
15 | 14 | ssrdv 3574 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹'''𝑥) ∈ 𝐵) → ran 𝐹 ⊆ 𝐵) |
16 | df-f 5808 | . . 3 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ran 𝐹 ⊆ 𝐵)) | |
17 | 5, 15, 16 | sylanbrc 695 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹'''𝑥) ∈ 𝐵) → 𝐹:𝐴⟶𝐵) |
18 | 4, 17 | impbii 198 | 1 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹'''𝑥) ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 ⊆ wss 3540 ran crn 5039 Fn wfn 5799 ⟶wf 5800 '''cafv 39843 |
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-res 5050 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fv 5812 df-dfat 39845 df-afv 39846 |
This theorem is referenced by: ffnaov 39928 |
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