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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  ontowfo 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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