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Theorem dmaf 16900
 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 (doma𝐴):𝐴𝐵

Proof of Theorem dmaf
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 fo1st 7353 . . . . . 6 1st :V–onto→V
2 fofn 6278 . . . . . 6 (1st :V–onto→V → 1st Fn V)
31, 2ax-mp 5 . . . . 5 1st Fn V
4 fof 6276 . . . . . 6 (1st :V–onto→V → 1st :V⟶V)
51, 4ax-mp 5 . . . . 5 1st :V⟶V
6 fnfco 6230 . . . . 5 ((1st Fn V ∧ 1st :V⟶V) → (1st ∘ 1st ) Fn V)
73, 5, 6mp2an 710 . . . 4 (1st ∘ 1st ) Fn V
8 df-doma 16875 . . . . 5 doma = (1st ∘ 1st )
98fneq1i 6146 . . . 4 (doma Fn V ↔ (1st ∘ 1st ) Fn V)
107, 9mpbir 221 . . 3 doma Fn V
11 ssv 3766 . . 3 𝐴 ⊆ V
12 fnssres 6165 . . 3 ((doma Fn V ∧ 𝐴 ⊆ V) → (doma𝐴) Fn 𝐴)
1310, 11, 12mp2an 710 . 2 (doma𝐴) Fn 𝐴
14 fvres 6368 . . . 4 (𝑥𝐴 → ((doma𝐴)‘𝑥) = (doma𝑥))
15 arwrcl.a . . . . 5 𝐴 = (Arrow‘𝐶)
16 arwdm.b . . . . 5 𝐵 = (Base‘𝐶)
1715, 16arwdm 16898 . . . 4 (𝑥𝐴 → (doma𝑥) ∈ 𝐵)
1814, 17eqeltrd 2839 . . 3 (𝑥𝐴 → ((doma𝐴)‘𝑥) ∈ 𝐵)
1918rgen 3060 . 2 𝑥𝐴 ((doma𝐴)‘𝑥) ∈ 𝐵
20 ffnfv 6551 . 2 ((doma𝐴):𝐴𝐵 ↔ ((doma𝐴) Fn 𝐴 ∧ ∀𝑥𝐴 ((doma𝐴)‘𝑥) ∈ 𝐵))
2113, 19, 20mpbir2an 993 1 (doma𝐴):𝐴𝐵
 Colors of variables: wff setvar class Syntax hints:   = wceq 1632   ∈ wcel 2139  ∀wral 3050  Vcvv 3340   ⊆ wss 3715   ↾ cres 5268   ∘ ccom 5270   Fn wfn 6044  ⟶wf 6045  –onto→wfo 6047  ‘cfv 6049  1st c1st 7331  Basecbs 16059  domacdoma 16871  Arrowcarw 16873 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6816  df-1st 7333  df-2nd 7334  df-doma 16875  df-coda 16876  df-homa 16877  df-arw 16878 This theorem is referenced by: (None)
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