Theorem dmaf 18102

Description: The domain function is a function from arrows to objects. (Contributed by Mario Carneiro, 11-Jan-2017.)

Hypotheses

Ref	Expression
arwrcl.a	⊢ 𝐴 = (Arrow‘𝐶)
arwdm.b	⊢ 𝐵 = (Base‘𝐶)

Assertion

Ref	Expression
dmaf	⊢ (doma ↾ 𝐴):𝐴⟶𝐵

Proof of Theorem dmaf

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fo1st 8002	. . . . . 6 ⊢ 1st :V–onto→V
2		fofn 6792	. . . . . 6 ⊢ (1st :V–onto→V → 1st Fn V)
3	1, 2	ax-mp 5	. . . . 5 ⊢ 1st Fn V
4		fof 6790	. . . . . 6 ⊢ (1st :V–onto→V → 1st :V⟶V)
5	1, 4	ax-mp 5	. . . . 5 ⊢ 1st :V⟶V
6		fnfco 6741	. . . . 5 ⊢ ((1st Fn V ∧ 1st :V⟶V) → (1st ∘ 1st ) Fn V)
7	3, 5, 6	mp2an 704	. . . 4 ⊢ (1st ∘ 1st ) Fn V
8		df-doma 18077	. . . . 5 ⊢ doma = (1st ∘ 1st )
9	8	fneq1i 6630	. . . 4 ⊢ (doma Fn V ↔ (1st ∘ 1st ) Fn V)
10	7, 9	mpbir 234	. . 3 ⊢ doma Fn V
11		ssv 3969	. . 3 ⊢ 𝐴 ⊆ V
12		fnssres 6656	. . 3 ⊢ ((doma Fn V ∧ 𝐴 ⊆ V) → (doma ↾ 𝐴) Fn 𝐴)
13	10, 11, 12	mp2an 704	. 2 ⊢ (doma ↾ 𝐴) Fn 𝐴
14		fvres 6898	. . . 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 18100	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (doma‘𝑥) ∈ 𝐵)
18	14, 17	eqeltrd 2869	. . 3 ⊢ (𝑥 ∈ 𝐴 → ((doma ↾ 𝐴)‘𝑥) ∈ 𝐵)
19	18	rgen 3087	. 2 ⊢ ∀𝑥 ∈ 𝐴 ((doma ↾ 𝐴)‘𝑥) ∈ 𝐵
20		ffnfv 7112	. 2 ⊢ ((doma ↾ 𝐴):𝐴⟶𝐵 ↔ ((doma ↾ 𝐴) Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 ((doma ↾ 𝐴)‘𝑥) ∈ 𝐵))
21	13, 19, 20	mpbir2an 723	1 ⊢ (doma ↾ 𝐴):𝐴⟶𝐵

Syntax hints:   = wceq 1567   ∈ wcel 2149  ∀wral 3085  Vcvv 3463   ⊆ wss 3913   ↾ cres 5661   ∘ ccom 5663   Fn wfn 6528  ⟶wf 6529  –onto→wfo 6531  ‘cfv 6533  1^st c1st 7980  Basecbs 17265  dom_acdoma 18073  Arrowcarw 18075

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-rep 5239  ax-sep 5258  ax-nul 5268  ax-pow 5334  ax-pr 5402  ax-un 7730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-ov 7411  df-1st 7982  df-2nd 7983  df-doma 18077  df-coda 18078  df-homa 18079  df-arw 18080

This theorem is referenced by: (None)