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Mirrors > Home > MPE Home > Th. List > Mathboxes > funimass4f | Structured version Visualization version GIF version |
Description: Membership relation for the values of a function whose image is a subclass. (Contributed by Thierry Arnoux, 24-Apr-2017.) |
Ref | Expression |
---|---|
funimass4f.1 | ⊢ Ⅎ𝑥𝐴 |
funimass4f.2 | ⊢ Ⅎ𝑥𝐵 |
funimass4f.3 | ⊢ Ⅎ𝑥𝐹 |
Ref | Expression |
---|---|
funimass4f | ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ((𝐹 “ 𝐴) ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funimass4f.3 | . . . . . 6 ⊢ Ⅎ𝑥𝐹 | |
2 | 1 | nffun 5826 | . . . . 5 ⊢ Ⅎ𝑥Fun 𝐹 |
3 | funimass4f.1 | . . . . . 6 ⊢ Ⅎ𝑥𝐴 | |
4 | 1 | nfdm 5288 | . . . . . 6 ⊢ Ⅎ𝑥dom 𝐹 |
5 | 3, 4 | nfss 3561 | . . . . 5 ⊢ Ⅎ𝑥 𝐴 ⊆ dom 𝐹 |
6 | 2, 5 | nfan 1816 | . . . 4 ⊢ Ⅎ𝑥(Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) |
7 | 1, 3 | nfima 5393 | . . . . 5 ⊢ Ⅎ𝑥(𝐹 “ 𝐴) |
8 | funimass4f.2 | . . . . 5 ⊢ Ⅎ𝑥𝐵 | |
9 | 7, 8 | nfss 3561 | . . . 4 ⊢ Ⅎ𝑥(𝐹 “ 𝐴) ⊆ 𝐵 |
10 | 6, 9 | nfan 1816 | . . 3 ⊢ Ⅎ𝑥((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝐹 “ 𝐴) ⊆ 𝐵) |
11 | funfvima2 6397 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝑥 ∈ 𝐴 → (𝐹‘𝑥) ∈ (𝐹 “ 𝐴))) | |
12 | ssel 3562 | . . . 4 ⊢ ((𝐹 “ 𝐴) ⊆ 𝐵 → ((𝐹‘𝑥) ∈ (𝐹 “ 𝐴) → (𝐹‘𝑥) ∈ 𝐵)) | |
13 | 11, 12 | sylan9 687 | . . 3 ⊢ (((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝐹 “ 𝐴) ⊆ 𝐵) → (𝑥 ∈ 𝐴 → (𝐹‘𝑥) ∈ 𝐵)) |
14 | 10, 13 | ralrimi 2940 | . 2 ⊢ (((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) ∧ (𝐹 “ 𝐴) ⊆ 𝐵) → ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵) |
15 | 3, 1 | dfimafnf 28817 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)}) |
16 | 15 | adantr 480 | . . 3 ⊢ (((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵) → (𝐹 “ 𝐴) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)}) |
17 | 8 | abrexss 28734 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵 → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)} ⊆ 𝐵) |
18 | 17 | adantl 481 | . . 3 ⊢ (((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵) → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = (𝐹‘𝑥)} ⊆ 𝐵) |
19 | 16, 18 | eqsstrd 3602 | . 2 ⊢ (((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵) → (𝐹 “ 𝐴) ⊆ 𝐵) |
20 | 14, 19 | impbida 873 | 1 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → ((𝐹 “ 𝐴) ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 {cab 2596 Ⅎwnfc 2738 ∀wral 2896 ∃wrex 2897 ⊆ wss 3540 dom cdm 5038 “ cima 5041 Fun wfun 5798 ‘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-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-fv 5812 |
This theorem is referenced by: ballotlem7 29924 |
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