Theorem dmaf 18107

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 8007	. . . . . 6 ⊢ 1st :V–onto→V
2		fofn 6796	. . . . . 6 ⊢ (1st :V–onto→V → 1st Fn V)
3	1, 2	ax-mp 5	. . . . 5 ⊢ 1st Fn V
4		fof 6794	. . . . . 6 ⊢ (1st :V–onto→V → 1st :V⟶V)
5	1, 4	ax-mp 5	. . . . 5 ⊢ 1st :V⟶V
6		fnfco 6745	. . . . 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 18082	. . . . 5 ⊢ doma = (1st ∘ 1st )
9	8	fneq1i 6634	. . . 4 ⊢ (doma Fn V ↔ (1st ∘ 1st ) Fn V)
10	7, 9	mpbir 234	. . 3 ⊢ doma Fn V
11		ssv 3962	. . 3 ⊢ 𝐴 ⊆ V
12		fnssres 6660	. . 3 ⊢ ((doma Fn V ∧ 𝐴 ⊆ V) → (doma ↾ 𝐴) Fn 𝐴)
13	10, 11, 12	mp2an 704	. 2 ⊢ (doma ↾ 𝐴) Fn 𝐴
14		fvres 6902	. . . 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 18105	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (doma‘𝑥) ∈ 𝐵)
18	14, 17	eqeltrd 2863	. . 3 ⊢ (𝑥 ∈ 𝐴 → ((doma ↾ 𝐴)‘𝑥) ∈ 𝐵)
19	18	rgen 3081	. 2 ⊢ ∀𝑥 ∈ 𝐴 ((doma ↾ 𝐴)‘𝑥) ∈ 𝐵
20		ffnfv 7116	. 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 1570   ∈ wcel 2143  ∀wral 3079  Vcvv 3455   ⊆ wss 3906   ↾ cres 5665   ∘ ccom 5667   Fn wfn 6533  ⟶wf 6534  –onto→wfo 6536  ‘cfv 6538  1^st c1st 7985  Basecbs 17270  dom_acdoma 18078  Arrowcarw 18080

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5239  ax-sep 5258  ax-nul 5270  ax-pow 5338  ax-pr 5406  ax-un 7734

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-iun 4959  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7415  df-1st 7987  df-2nd 7988  df-doma 18082  df-coda 18083  df-homa 18084  df-arw 18085

This theorem is referenced by: (None)