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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 8034 | . . . . . 6 ⊢ 1st :V–onto→V | |
| 2 | fofn 6822 | . . . . . 6 ⊢ (1st :V–onto→V → 1st Fn V) | |
| 3 | 1, 2 | ax-mp 5 | . . . . 5 ⊢ 1st Fn V |
| 4 | fof 6820 | . . . . . 6 ⊢ (1st :V–onto→V → 1st :V⟶V) | |
| 5 | 1, 4 | ax-mp 5 | . . . . 5 ⊢ 1st :V⟶V |
| 6 | fnfco 6773 | . . . . 5 ⊢ ((1st Fn V ∧ 1st :V⟶V) → (1st ∘ 1st ) Fn V) | |
| 7 | 3, 5, 6 | mp2an 692 | . . . 4 ⊢ (1st ∘ 1st ) Fn V |
| 8 | df-doma 18069 | . . . . 5 ⊢ doma = (1st ∘ 1st ) | |
| 9 | 8 | fneq1i 6665 | . . . 4 ⊢ (doma Fn V ↔ (1st ∘ 1st ) Fn V) |
| 10 | 7, 9 | mpbir 231 | . . 3 ⊢ doma Fn V |
| 11 | ssv 4008 | . . 3 ⊢ 𝐴 ⊆ V | |
| 12 | fnssres 6691 | . . 3 ⊢ ((doma Fn V ∧ 𝐴 ⊆ V) → (doma ↾ 𝐴) Fn 𝐴) | |
| 13 | 10, 11, 12 | mp2an 692 | . 2 ⊢ (doma ↾ 𝐴) Fn 𝐴 |
| 14 | fvres 6925 | . . . 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 18092 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → (doma‘𝑥) ∈ 𝐵) |
| 18 | 14, 17 | eqeltrd 2841 | . . 3 ⊢ (𝑥 ∈ 𝐴 → ((doma ↾ 𝐴)‘𝑥) ∈ 𝐵) |
| 19 | 18 | rgen 3063 | . 2 ⊢ ∀𝑥 ∈ 𝐴 ((doma ↾ 𝐴)‘𝑥) ∈ 𝐵 |
| 20 | ffnfv 7139 | . 2 ⊢ ((doma ↾ 𝐴):𝐴⟶𝐵 ↔ ((doma ↾ 𝐴) Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 ((doma ↾ 𝐴)‘𝑥) ∈ 𝐵)) | |
| 21 | 13, 19, 20 | mpbir2an 711 | 1 ⊢ (doma ↾ 𝐴):𝐴⟶𝐵 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∈ wcel 2108 ∀wral 3061 Vcvv 3480 ⊆ wss 3951 ↾ cres 5687 ∘ ccom 5689 Fn wfn 6556 ⟶wf 6557 –onto→wfo 6559 ‘cfv 6561 1st c1st 8012 Basecbs 17247 domacdoma 18065 Arrowcarw 18067 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-1st 8014 df-2nd 8015 df-doma 18069 df-coda 18070 df-homa 18071 df-arw 18072 |
| This theorem is referenced by: (None) |
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