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Mirrors > Home > MPE Home > Th. List > dmaf | Structured version Visualization version GIF version |
Description: The domain function is a function from arrows to objects. (Contributed by Mario Carneiro, 11-Jan-2017.) |
Ref | Expression |
---|---|
arwrcl.a | ⊢ 𝐴 = (Arrow‘𝐶) |
arwdm.b | ⊢ 𝐵 = (Base‘𝐶) |
Ref | Expression |
---|---|
dmaf | ⊢ (doma ↾ 𝐴):𝐴⟶𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fo1st 7230 | . . . . . 6 ⊢ 1st :V–onto→V | |
2 | fofn 6155 | . . . . . 6 ⊢ (1st :V–onto→V → 1st Fn V) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 ⊢ 1st Fn V |
4 | fof 6153 | . . . . . 6 ⊢ (1st :V–onto→V → 1st :V⟶V) | |
5 | 1, 4 | ax-mp 5 | . . . . 5 ⊢ 1st :V⟶V |
6 | fnfco 6107 | . . . . 5 ⊢ ((1st Fn V ∧ 1st :V⟶V) → (1st ∘ 1st ) Fn V) | |
7 | 3, 5, 6 | mp2an 708 | . . . 4 ⊢ (1st ∘ 1st ) Fn V |
8 | df-doma 16721 | . . . . 5 ⊢ doma = (1st ∘ 1st ) | |
9 | 8 | fneq1i 6023 | . . . 4 ⊢ (doma Fn V ↔ (1st ∘ 1st ) Fn V) |
10 | 7, 9 | mpbir 221 | . . 3 ⊢ doma Fn V |
11 | ssv 3658 | . . 3 ⊢ 𝐴 ⊆ V | |
12 | fnssres 6042 | . . 3 ⊢ ((doma Fn V ∧ 𝐴 ⊆ V) → (doma ↾ 𝐴) Fn 𝐴) | |
13 | 10, 11, 12 | mp2an 708 | . 2 ⊢ (doma ↾ 𝐴) Fn 𝐴 |
14 | fvres 6245 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ((doma ↾ 𝐴)‘𝑥) = (doma‘𝑥)) | |
15 | arwrcl.a | . . . . 5 ⊢ 𝐴 = (Arrow‘𝐶) | |
16 | arwdm.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐶) | |
17 | 15, 16 | arwdm 16744 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → (doma‘𝑥) ∈ 𝐵) |
18 | 14, 17 | eqeltrd 2730 | . . 3 ⊢ (𝑥 ∈ 𝐴 → ((doma ↾ 𝐴)‘𝑥) ∈ 𝐵) |
19 | 18 | rgen 2951 | . 2 ⊢ ∀𝑥 ∈ 𝐴 ((doma ↾ 𝐴)‘𝑥) ∈ 𝐵 |
20 | ffnfv 6428 | . 2 ⊢ ((doma ↾ 𝐴):𝐴⟶𝐵 ↔ ((doma ↾ 𝐴) Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 ((doma ↾ 𝐴)‘𝑥) ∈ 𝐵)) | |
21 | 13, 19, 20 | mpbir2an 975 | 1 ⊢ (doma ↾ 𝐴):𝐴⟶𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1523 ∈ wcel 2030 ∀wral 2941 Vcvv 3231 ⊆ wss 3607 ↾ cres 5145 ∘ ccom 5147 Fn wfn 5921 ⟶wf 5922 –onto→wfo 5924 ‘cfv 5926 1st c1st 7208 Basecbs 15904 domacdoma 16717 Arrowcarw 16719 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-op 4217 df-uni 4469 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-id 5053 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-ov 6693 df-1st 7210 df-2nd 7211 df-doma 16721 df-coda 16722 df-homa 16723 df-arw 16724 |
This theorem is referenced by: (None) |
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