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Mirrors > Home > MPE Home > Th. List > cdaf | Structured version Visualization version GIF version |
Description: The codomain 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 |
---|---|
cdaf | ⊢ (coda ↾ 𝐴):𝐴⟶𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fo2nd 7842 | . . . . . 6 ⊢ 2nd :V–onto→V | |
2 | fofn 6683 | . . . . . 6 ⊢ (2nd :V–onto→V → 2nd Fn V) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 ⊢ 2nd Fn V |
4 | fo1st 7841 | . . . . . 6 ⊢ 1st :V–onto→V | |
5 | fof 6681 | . . . . . 6 ⊢ (1st :V–onto→V → 1st :V⟶V) | |
6 | 4, 5 | ax-mp 5 | . . . . 5 ⊢ 1st :V⟶V |
7 | fnfco 6632 | . . . . 5 ⊢ ((2nd Fn V ∧ 1st :V⟶V) → (2nd ∘ 1st ) Fn V) | |
8 | 3, 6, 7 | mp2an 689 | . . . 4 ⊢ (2nd ∘ 1st ) Fn V |
9 | df-coda 17728 | . . . . 5 ⊢ coda = (2nd ∘ 1st ) | |
10 | 9 | fneq1i 6523 | . . . 4 ⊢ (coda Fn V ↔ (2nd ∘ 1st ) Fn V) |
11 | 8, 10 | mpbir 230 | . . 3 ⊢ coda Fn V |
12 | ssv 3945 | . . 3 ⊢ 𝐴 ⊆ V | |
13 | fnssres 6548 | . . 3 ⊢ ((coda Fn V ∧ 𝐴 ⊆ V) → (coda ↾ 𝐴) Fn 𝐴) | |
14 | 11, 12, 13 | mp2an 689 | . 2 ⊢ (coda ↾ 𝐴) Fn 𝐴 |
15 | fvres 6786 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → ((coda ↾ 𝐴)‘𝑥) = (coda‘𝑥)) | |
16 | arwrcl.a | . . . . 5 ⊢ 𝐴 = (Arrow‘𝐶) | |
17 | arwdm.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐶) | |
18 | 16, 17 | arwcd 17751 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → (coda‘𝑥) ∈ 𝐵) |
19 | 15, 18 | eqeltrd 2839 | . . 3 ⊢ (𝑥 ∈ 𝐴 → ((coda ↾ 𝐴)‘𝑥) ∈ 𝐵) |
20 | 19 | rgen 3074 | . 2 ⊢ ∀𝑥 ∈ 𝐴 ((coda ↾ 𝐴)‘𝑥) ∈ 𝐵 |
21 | ffnfv 6985 | . 2 ⊢ ((coda ↾ 𝐴):𝐴⟶𝐵 ↔ ((coda ↾ 𝐴) Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 ((coda ↾ 𝐴)‘𝑥) ∈ 𝐵)) | |
22 | 14, 20, 21 | mpbir2an 708 | 1 ⊢ (coda ↾ 𝐴):𝐴⟶𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1539 ∈ wcel 2106 ∀wral 3064 Vcvv 3430 ⊆ wss 3887 ↾ cres 5587 ∘ ccom 5589 Fn wfn 6422 ⟶wf 6423 –onto→wfo 6425 ‘cfv 6427 1st c1st 7819 2nd c2nd 7820 Basecbs 16900 codaccoda 17724 Arrowcarw 17725 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5222 ax-nul 5229 ax-pow 5287 ax-pr 5351 ax-un 7579 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3071 df-rab 3073 df-v 3432 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4258 df-if 4461 df-pw 4536 df-sn 4563 df-pr 4565 df-op 4569 df-uni 4841 df-iun 4927 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5485 df-xp 5591 df-rel 5592 df-cnv 5593 df-co 5594 df-dm 5595 df-rn 5596 df-res 5597 df-ima 5598 df-iota 6385 df-fun 6429 df-fn 6430 df-f 6431 df-f1 6432 df-fo 6433 df-f1o 6434 df-fv 6435 df-ov 7271 df-1st 7821 df-2nd 7822 df-doma 17727 df-coda 17728 df-homa 17729 df-arw 17730 |
This theorem is referenced by: (None) |
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