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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 7844 | . . . . . 6 ⊢ 1st :V–onto→V | |
2 | fofn 6688 | . . . . . 6 ⊢ (1st :V–onto→V → 1st Fn V) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 ⊢ 1st Fn V |
4 | fof 6686 | . . . . . 6 ⊢ (1st :V–onto→V → 1st :V⟶V) | |
5 | 1, 4 | ax-mp 5 | . . . . 5 ⊢ 1st :V⟶V |
6 | fnfco 6637 | . . . . 5 ⊢ ((1st Fn V ∧ 1st :V⟶V) → (1st ∘ 1st ) Fn V) | |
7 | 3, 5, 6 | mp2an 689 | . . . 4 ⊢ (1st ∘ 1st ) Fn V |
8 | df-doma 17737 | . . . . 5 ⊢ doma = (1st ∘ 1st ) | |
9 | 8 | fneq1i 6528 | . . . 4 ⊢ (doma Fn V ↔ (1st ∘ 1st ) Fn V) |
10 | 7, 9 | mpbir 230 | . . 3 ⊢ doma Fn V |
11 | ssv 3950 | . . 3 ⊢ 𝐴 ⊆ V | |
12 | fnssres 6553 | . . 3 ⊢ ((doma Fn V ∧ 𝐴 ⊆ V) → (doma ↾ 𝐴) Fn 𝐴) | |
13 | 10, 11, 12 | mp2an 689 | . 2 ⊢ (doma ↾ 𝐴) Fn 𝐴 |
14 | fvres 6790 | . . . 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 17760 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → (doma‘𝑥) ∈ 𝐵) |
18 | 14, 17 | eqeltrd 2841 | . . 3 ⊢ (𝑥 ∈ 𝐴 → ((doma ↾ 𝐴)‘𝑥) ∈ 𝐵) |
19 | 18 | rgen 3076 | . 2 ⊢ ∀𝑥 ∈ 𝐴 ((doma ↾ 𝐴)‘𝑥) ∈ 𝐵 |
20 | ffnfv 6989 | . 2 ⊢ ((doma ↾ 𝐴):𝐴⟶𝐵 ↔ ((doma ↾ 𝐴) Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 ((doma ↾ 𝐴)‘𝑥) ∈ 𝐵)) | |
21 | 13, 19, 20 | mpbir2an 708 | 1 ⊢ (doma ↾ 𝐴):𝐴⟶𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2110 ∀wral 3066 Vcvv 3431 ⊆ wss 3892 ↾ cres 5592 ∘ ccom 5594 Fn wfn 6427 ⟶wf 6428 –onto→wfo 6430 ‘cfv 6432 1st c1st 7822 Basecbs 16910 domacdoma 17733 Arrowcarw 17735 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2015 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2711 ax-rep 5214 ax-sep 5227 ax-nul 5234 ax-pow 5292 ax-pr 5356 ax-un 7582 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2072 df-mo 2542 df-eu 2571 df-clab 2718 df-cleq 2732 df-clel 2818 df-nfc 2891 df-ne 2946 df-ral 3071 df-rex 3072 df-reu 3073 df-rab 3075 df-v 3433 df-sbc 3721 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4846 df-iun 4932 df-br 5080 df-opab 5142 df-mpt 5163 df-id 5490 df-xp 5596 df-rel 5597 df-cnv 5598 df-co 5599 df-dm 5600 df-rn 5601 df-res 5602 df-ima 5603 df-iota 6390 df-fun 6434 df-fn 6435 df-f 6436 df-f1 6437 df-fo 6438 df-f1o 6439 df-fv 6440 df-ov 7274 df-1st 7824 df-2nd 7825 df-doma 17737 df-coda 17738 df-homa 17739 df-arw 17740 |
This theorem is referenced by: (None) |
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