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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 7691 | . . . . . 6 ⊢ 1st :V–onto→V | |
2 | fofn 6567 | . . . . . 6 ⊢ (1st :V–onto→V → 1st Fn V) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 ⊢ 1st Fn V |
4 | fof 6565 | . . . . . 6 ⊢ (1st :V–onto→V → 1st :V⟶V) | |
5 | 1, 4 | ax-mp 5 | . . . . 5 ⊢ 1st :V⟶V |
6 | fnfco 6517 | . . . . 5 ⊢ ((1st Fn V ∧ 1st :V⟶V) → (1st ∘ 1st ) Fn V) | |
7 | 3, 5, 6 | mp2an 691 | . . . 4 ⊢ (1st ∘ 1st ) Fn V |
8 | df-doma 17276 | . . . . 5 ⊢ doma = (1st ∘ 1st ) | |
9 | 8 | fneq1i 6420 | . . . 4 ⊢ (doma Fn V ↔ (1st ∘ 1st ) Fn V) |
10 | 7, 9 | mpbir 234 | . . 3 ⊢ doma Fn V |
11 | ssv 3939 | . . 3 ⊢ 𝐴 ⊆ V | |
12 | fnssres 6442 | . . 3 ⊢ ((doma Fn V ∧ 𝐴 ⊆ V) → (doma ↾ 𝐴) Fn 𝐴) | |
13 | 10, 11, 12 | mp2an 691 | . 2 ⊢ (doma ↾ 𝐴) Fn 𝐴 |
14 | fvres 6664 | . . . 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 17299 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → (doma‘𝑥) ∈ 𝐵) |
18 | 14, 17 | eqeltrd 2890 | . . 3 ⊢ (𝑥 ∈ 𝐴 → ((doma ↾ 𝐴)‘𝑥) ∈ 𝐵) |
19 | 18 | rgen 3116 | . 2 ⊢ ∀𝑥 ∈ 𝐴 ((doma ↾ 𝐴)‘𝑥) ∈ 𝐵 |
20 | ffnfv 6859 | . 2 ⊢ ((doma ↾ 𝐴):𝐴⟶𝐵 ↔ ((doma ↾ 𝐴) Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 ((doma ↾ 𝐴)‘𝑥) ∈ 𝐵)) | |
21 | 13, 19, 20 | mpbir2an 710 | 1 ⊢ (doma ↾ 𝐴):𝐴⟶𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∈ wcel 2111 ∀wral 3106 Vcvv 3441 ⊆ wss 3881 ↾ cres 5521 ∘ ccom 5523 Fn wfn 6319 ⟶wf 6320 –onto→wfo 6322 ‘cfv 6324 1st c1st 7669 Basecbs 16475 domacdoma 17272 Arrowcarw 17274 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-1st 7671 df-2nd 7672 df-doma 17276 df-coda 17277 df-homa 17278 df-arw 17279 |
This theorem is referenced by: (None) |
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