Theorem dmaf 18274

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 8055	. . . . . 6 ⊢ 1st :V⟶ontoV
2		fofn 6828	. . . . . 6 ⊢ (1st :V⟶ontoV → 1st Fn V)
3	1, 2	ax-mp 5	. . . . 5 ⊢ 1st Fn V
4		fof 6826	. . . . . 6 ⊢ (1st :V⟶ontoV → 1st :V⟶V)
5	1, 4	ax-mp 5	. . . . 5 ⊢ 1st :V⟶V
6		fnfco 6777	. . . . 5 ⊢ ((1st Fn V ∧ 1st :V⟶V) → (1st ∘ 1st ) Fn V)
7	3, 5, 6	mp2an 705	. . . 4 ⊢ (1st ∘ 1st ) Fn V
8		df-doma 18249	. . . . 5 ⊢ doma = (1st ∘ 1st )
9	8	fneq1i 6666	. . . 4 ⊢ (doma Fn V ↔ (1st ∘ 1st ) Fn V)
10	7, 9	mpbir 234	. . 3 ⊢ doma Fn V
11		ssv 3937	. . 3 ⊢ 𝐴 ⊆ V
12		fnssres 6692	. . 3 ⊢ ((doma Fn V ∧ 𝐴 ⊆ V) → (doma ↾ 𝐴) Fn 𝐴)
13	10, 11, 12	mp2an 705	. 2 ⊢ (doma ↾ 𝐴) Fn 𝐴
14		fvres 6935	. . . 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 18272	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (doma‘𝑥) ∈ 𝐵)
18	14, 17	eqeltrd 2864	. . 3 ⊢ (𝑥 ∈ 𝐴 → ((doma ↾ 𝐴)‘𝑥) ∈ 𝐵)
19	18	rgen 3082	. 2 ⊢ ∀𝑥 ∈ 𝐴 ((doma ↾ 𝐴)‘𝑥) ∈ 𝐵
20		ffnfv 7151	. 2 ⊢ ((doma ↾ 𝐴):𝐴⟶𝐵 ↔ ((doma ↾ 𝐴) Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 ((doma ↾ 𝐴)‘𝑥) ∈ 𝐵))
21	13, 19, 20	mpbir2an 724	1 ⊢ (doma ↾ 𝐴):𝐴⟶𝐵

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2147  ∀wral 3080  Vcvv 3454   ⊆ wss 3900   ↾ cres 5660   ∘ ccom 5662   Fn wfn 6561  ⟶wf 6562  ⟶ontowfo 6564  ‘cfv 6566  1^st c1st 8033  Basecbs 17431  dom_acdoma 18245  Arrowcarw 18247

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-rep 5238  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-iun 4958  df-br 5109  df-opab 5174  df-mpt 5193  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-ov 7454  df-1st 8035  df-2nd 8036  df-doma 18249  df-coda 18250  df-homa 18251  df-arw 18252

This theorem is used by: (None)