Theorem dmaf 16699

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 7188	. . . . . 6 ⊢ 1st :V–onto→V
2		fofn 6117	. . . . . 6 ⊢ (1st :V–onto→V → 1st Fn V)
3	1, 2	ax-mp 5	. . . . 5 ⊢ 1st Fn V
4		fof 6115	. . . . . 6 ⊢ (1st :V–onto→V → 1st :V⟶V)
5	1, 4	ax-mp 5	. . . . 5 ⊢ 1st :V⟶V
6		fnfco 6069	. . . . 5 ⊢ ((1st Fn V ∧ 1st :V⟶V) → (1st ∘ 1st ) Fn V)
7	3, 5, 6	mp2an 708	. . . 4 ⊢ (1st ∘ 1st ) Fn V
8		df-doma 16674	. . . . 5 ⊢ doma = (1st ∘ 1st )
9	8	fneq1i 5985	. . . 4 ⊢ (doma Fn V ↔ (1st ∘ 1st ) Fn V)
10	7, 9	mpbir 221	. . 3 ⊢ doma Fn V
11		ssv 3625	. . 3 ⊢ 𝐴 ⊆ V
12		fnssres 6004	. . 3 ⊢ ((doma Fn V ∧ 𝐴 ⊆ V) → (doma ↾ 𝐴) Fn 𝐴)
13	10, 11, 12	mp2an 708	. 2 ⊢ (doma ↾ 𝐴) Fn 𝐴
14		fvres 6207	. . . 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 16697	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (doma‘𝑥) ∈ 𝐵)
18	14, 17	eqeltrd 2701	. . 3 ⊢ (𝑥 ∈ 𝐴 → ((doma ↾ 𝐴)‘𝑥) ∈ 𝐵)
19	18	rgen 2922	. 2 ⊢ ∀𝑥 ∈ 𝐴 ((doma ↾ 𝐴)‘𝑥) ∈ 𝐵
20		ffnfv 6388	. 2 ⊢ ((doma ↾ 𝐴):𝐴⟶𝐵 ↔ ((doma ↾ 𝐴) Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 ((doma ↾ 𝐴)‘𝑥) ∈ 𝐵))
21	13, 19, 20	mpbir2an 955	1 ⊢ (doma ↾ 𝐴):𝐴⟶𝐵

Syntax hints:   = wceq 1483   ∈ wcel 1990  ∀wral 2912  Vcvv 3200   ⊆ wss 3574   ↾ cres 5116   ∘ ccom 5118   Fn wfn 5883  ⟶wf 5884  –onto→wfo 5886  ‘cfv 5888  1^st c1st 7166  Basecbs 15857  dom_acdoma 16670  Arrowcarw 16672

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-1st 7168  df-2nd 7169  df-doma 16674  df-coda 16675  df-homa 16676  df-arw 16677

This theorem is referenced by: (None)