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Theorem dmaf 18103
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 8033 . . . . . 6 1st :V–onto→V
2 fofn 6823 . . . . . 6 (1st :V–onto→V → 1st Fn V)
31, 2ax-mp 5 . . . . 5 1st Fn V
4 fof 6821 . . . . . 6 (1st :V–onto→V → 1st :V⟶V)
51, 4ax-mp 5 . . . . 5 1st :V⟶V
6 fnfco 6774 . . . . 5 ((1st Fn V ∧ 1st :V⟶V) → (1st ∘ 1st ) Fn V)
73, 5, 6mp2an 692 . . . 4 (1st ∘ 1st ) Fn V
8 df-doma 18078 . . . . 5 doma = (1st ∘ 1st )
98fneq1i 6666 . . . 4 (doma Fn V ↔ (1st ∘ 1st ) Fn V)
107, 9mpbir 231 . . 3 doma Fn V
11 ssv 4020 . . 3 𝐴 ⊆ V
12 fnssres 6692 . . 3 ((doma Fn V ∧ 𝐴 ⊆ V) → (doma𝐴) Fn 𝐴)
1310, 11, 12mp2an 692 . 2 (doma𝐴) Fn 𝐴
14 fvres 6926 . . . 4 (𝑥𝐴 → ((doma𝐴)‘𝑥) = (doma𝑥))
15 arwrcl.a . . . . 5 𝐴 = (Arrow‘𝐶)
16 arwdm.b . . . . 5 𝐵 = (Base‘𝐶)
1715, 16arwdm 18101 . . . 4 (𝑥𝐴 → (doma𝑥) ∈ 𝐵)
1814, 17eqeltrd 2839 . . 3 (𝑥𝐴 → ((doma𝐴)‘𝑥) ∈ 𝐵)
1918rgen 3061 . 2 𝑥𝐴 ((doma𝐴)‘𝑥) ∈ 𝐵
20 ffnfv 7139 . 2 ((doma𝐴):𝐴𝐵 ↔ ((doma𝐴) Fn 𝐴 ∧ ∀𝑥𝐴 ((doma𝐴)‘𝑥) ∈ 𝐵))
2113, 19, 20mpbir2an 711 1 (doma𝐴):𝐴𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wcel 2106  wral 3059  Vcvv 3478  wss 3963  cres 5691  ccom 5693   Fn wfn 6558  wf 6559  ontowfo 6561  cfv 6563  1st c1st 8011  Basecbs 17245  domacdoma 18074  Arrowcarw 18076
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-1st 8013  df-2nd 8014  df-doma 18078  df-coda 18079  df-homa 18080  df-arw 18081
This theorem is referenced by: (None)
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