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Theorem dmaf 17013
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 7421 . . . . . 6 1st :V–onto→V
2 fofn 6333 . . . . . 6 (1st :V–onto→V → 1st Fn V)
31, 2ax-mp 5 . . . . 5 1st Fn V
4 fof 6331 . . . . . 6 (1st :V–onto→V → 1st :V⟶V)
51, 4ax-mp 5 . . . . 5 1st :V⟶V
6 fnfco 6284 . . . . 5 ((1st Fn V ∧ 1st :V⟶V) → (1st ∘ 1st ) Fn V)
73, 5, 6mp2an 684 . . . 4 (1st ∘ 1st ) Fn V
8 df-doma 16988 . . . . 5 doma = (1st ∘ 1st )
98fneq1i 6196 . . . 4 (doma Fn V ↔ (1st ∘ 1st ) Fn V)
107, 9mpbir 223 . . 3 doma Fn V
11 ssv 3821 . . 3 𝐴 ⊆ V
12 fnssres 6215 . . 3 ((doma Fn V ∧ 𝐴 ⊆ V) → (doma𝐴) Fn 𝐴)
1310, 11, 12mp2an 684 . 2 (doma𝐴) Fn 𝐴
14 fvres 6430 . . . 4 (𝑥𝐴 → ((doma𝐴)‘𝑥) = (doma𝑥))
15 arwrcl.a . . . . 5 𝐴 = (Arrow‘𝐶)
16 arwdm.b . . . . 5 𝐵 = (Base‘𝐶)
1715, 16arwdm 17011 . . . 4 (𝑥𝐴 → (doma𝑥) ∈ 𝐵)
1814, 17eqeltrd 2878 . . 3 (𝑥𝐴 → ((doma𝐴)‘𝑥) ∈ 𝐵)
1918rgen 3103 . 2 𝑥𝐴 ((doma𝐴)‘𝑥) ∈ 𝐵
20 ffnfv 6614 . 2 ((doma𝐴):𝐴𝐵 ↔ ((doma𝐴) Fn 𝐴 ∧ ∀𝑥𝐴 ((doma𝐴)‘𝑥) ∈ 𝐵))
2113, 19, 20mpbir2an 703 1 (doma𝐴):𝐴𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1653  wcel 2157  wral 3089  Vcvv 3385  wss 3769  cres 5314  ccom 5316   Fn wfn 6096  wf 6097  ontowfo 6099  cfv 6101  1st c1st 7399  Basecbs 16184  domacdoma 16984  Arrowcarw 16986
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-ov 6881  df-1st 7401  df-2nd 7402  df-doma 16988  df-coda 16989  df-homa 16990  df-arw 16991
This theorem is referenced by: (None)
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