Theorem dmaf 18007

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 7955	. . . . . 6 ⊢ 1st :V–onto→V
2		fofn 6748	. . . . . 6 ⊢ (1st :V–onto→V → 1st Fn V)
3	1, 2	ax-mp 5	. . . . 5 ⊢ 1st Fn V
4		fof 6746	. . . . . 6 ⊢ (1st :V–onto→V → 1st :V⟶V)
5	1, 4	ax-mp 5	. . . . 5 ⊢ 1st :V⟶V
6		fnfco 6699	. . . . 5 ⊢ ((1st Fn V ∧ 1st :V⟶V) → (1st ∘ 1st ) Fn V)
7	3, 5, 6	mp2an 693	. . . 4 ⊢ (1st ∘ 1st ) Fn V
8		df-doma 17982	. . . . 5 ⊢ doma = (1st ∘ 1st )
9	8	fneq1i 6589	. . . 4 ⊢ (doma Fn V ↔ (1st ∘ 1st ) Fn V)
10	7, 9	mpbir 231	. . 3 ⊢ doma Fn V
11		ssv 3947	. . 3 ⊢ 𝐴 ⊆ V
12		fnssres 6615	. . 3 ⊢ ((doma Fn V ∧ 𝐴 ⊆ V) → (doma ↾ 𝐴) Fn 𝐴)
13	10, 11, 12	mp2an 693	. 2 ⊢ (doma ↾ 𝐴) Fn 𝐴
14		fvres 6853	. . . 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 18005	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (doma‘𝑥) ∈ 𝐵)
18	14, 17	eqeltrd 2837	. . 3 ⊢ (𝑥 ∈ 𝐴 → ((doma ↾ 𝐴)‘𝑥) ∈ 𝐵)
19	18	rgen 3054	. 2 ⊢ ∀𝑥 ∈ 𝐴 ((doma ↾ 𝐴)‘𝑥) ∈ 𝐵
20		ffnfv 7065	. 2 ⊢ ((doma ↾ 𝐴):𝐴⟶𝐵 ↔ ((doma ↾ 𝐴) Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 ((doma ↾ 𝐴)‘𝑥) ∈ 𝐵))
21	13, 19, 20	mpbir2an 712	1 ⊢ (doma ↾ 𝐴):𝐴⟶𝐵

Syntax hints:   = wceq 1542   ∈ wcel 2114  ∀wral 3052  Vcvv 3430   ⊆ wss 3890   ↾ cres 5626   ∘ ccom 5628   Fn wfn 6487  ⟶wf 6488  –onto→wfo 6490  ‘cfv 6492  1^st c1st 7933  Basecbs 17170  dom_acdoma 17978  Arrowcarw 17980

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7363  df-1st 7935  df-2nd 7936  df-doma 17982  df-coda 17983  df-homa 17984  df-arw 17985

This theorem is referenced by: (None)