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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 7698 | . . . . . 6 ⊢ 1st :V–onto→V | |
2 | fofn 6585 | . . . . . 6 ⊢ (1st :V–onto→V → 1st Fn V) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 ⊢ 1st Fn V |
4 | fof 6583 | . . . . . 6 ⊢ (1st :V–onto→V → 1st :V⟶V) | |
5 | 1, 4 | ax-mp 5 | . . . . 5 ⊢ 1st :V⟶V |
6 | fnfco 6536 | . . . . 5 ⊢ ((1st Fn V ∧ 1st :V⟶V) → (1st ∘ 1st ) Fn V) | |
7 | 3, 5, 6 | mp2an 688 | . . . 4 ⊢ (1st ∘ 1st ) Fn V |
8 | df-doma 17272 | . . . . 5 ⊢ doma = (1st ∘ 1st ) | |
9 | 8 | fneq1i 6443 | . . . 4 ⊢ (doma Fn V ↔ (1st ∘ 1st ) Fn V) |
10 | 7, 9 | mpbir 232 | . . 3 ⊢ doma Fn V |
11 | ssv 3988 | . . 3 ⊢ 𝐴 ⊆ V | |
12 | fnssres 6463 | . . 3 ⊢ ((doma Fn V ∧ 𝐴 ⊆ V) → (doma ↾ 𝐴) Fn 𝐴) | |
13 | 10, 11, 12 | mp2an 688 | . 2 ⊢ (doma ↾ 𝐴) Fn 𝐴 |
14 | fvres 6682 | . . . 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 17295 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → (doma‘𝑥) ∈ 𝐵) |
18 | 14, 17 | eqeltrd 2910 | . . 3 ⊢ (𝑥 ∈ 𝐴 → ((doma ↾ 𝐴)‘𝑥) ∈ 𝐵) |
19 | 18 | rgen 3145 | . 2 ⊢ ∀𝑥 ∈ 𝐴 ((doma ↾ 𝐴)‘𝑥) ∈ 𝐵 |
20 | ffnfv 6874 | . 2 ⊢ ((doma ↾ 𝐴):𝐴⟶𝐵 ↔ ((doma ↾ 𝐴) Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 ((doma ↾ 𝐴)‘𝑥) ∈ 𝐵)) | |
21 | 13, 19, 20 | mpbir2an 707 | 1 ⊢ (doma ↾ 𝐴):𝐴⟶𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∈ wcel 2105 ∀wral 3135 Vcvv 3492 ⊆ wss 3933 ↾ cres 5550 ∘ ccom 5552 Fn wfn 6343 ⟶wf 6344 –onto→wfo 6346 ‘cfv 6348 1st c1st 7676 Basecbs 16471 domacdoma 17268 Arrowcarw 17270 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-1st 7678 df-2nd 7679 df-doma 17272 df-coda 17273 df-homa 17274 df-arw 17275 |
This theorem is referenced by: (None) |
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