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Theorem dmaf 18003
Description: The domain function is a function from arrows to objects. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
arwrcl.a 𝐴 = (Arrowβ€˜πΆ)
arwdm.b 𝐡 = (Baseβ€˜πΆ)
Assertion
Ref Expression
dmaf (doma β†Ύ 𝐴):𝐴⟢𝐡

Proof of Theorem dmaf
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 fo1st 7989 . . . . . 6 1st :V–ontoβ†’V
2 fofn 6798 . . . . . 6 (1st :V–ontoβ†’V β†’ 1st Fn V)
31, 2ax-mp 5 . . . . 5 1st Fn V
4 fof 6796 . . . . . 6 (1st :V–ontoβ†’V β†’ 1st :V⟢V)
51, 4ax-mp 5 . . . . 5 1st :V⟢V
6 fnfco 6747 . . . . 5 ((1st Fn V ∧ 1st :V⟢V) β†’ (1st ∘ 1st ) Fn V)
73, 5, 6mp2an 689 . . . 4 (1st ∘ 1st ) Fn V
8 df-doma 17978 . . . . 5 doma = (1st ∘ 1st )
98fneq1i 6637 . . . 4 (doma Fn V ↔ (1st ∘ 1st ) Fn V)
107, 9mpbir 230 . . 3 doma Fn V
11 ssv 3999 . . 3 𝐴 βŠ† V
12 fnssres 6664 . . 3 ((doma Fn V ∧ 𝐴 βŠ† V) β†’ (doma β†Ύ 𝐴) Fn 𝐴)
1310, 11, 12mp2an 689 . 2 (doma β†Ύ 𝐴) Fn 𝐴
14 fvres 6901 . . . 4 (π‘₯ ∈ 𝐴 β†’ ((doma β†Ύ 𝐴)β€˜π‘₯) = (domaβ€˜π‘₯))
15 arwrcl.a . . . . 5 𝐴 = (Arrowβ€˜πΆ)
16 arwdm.b . . . . 5 𝐡 = (Baseβ€˜πΆ)
1715, 16arwdm 18001 . . . 4 (π‘₯ ∈ 𝐴 β†’ (domaβ€˜π‘₯) ∈ 𝐡)
1814, 17eqeltrd 2825 . . 3 (π‘₯ ∈ 𝐴 β†’ ((doma β†Ύ 𝐴)β€˜π‘₯) ∈ 𝐡)
1918rgen 3055 . 2 βˆ€π‘₯ ∈ 𝐴 ((doma β†Ύ 𝐴)β€˜π‘₯) ∈ 𝐡
20 ffnfv 7111 . 2 ((doma β†Ύ 𝐴):𝐴⟢𝐡 ↔ ((doma β†Ύ 𝐴) Fn 𝐴 ∧ βˆ€π‘₯ ∈ 𝐴 ((doma β†Ύ 𝐴)β€˜π‘₯) ∈ 𝐡))
2113, 19, 20mpbir2an 708 1 (doma β†Ύ 𝐴):𝐴⟢𝐡
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533   ∈ wcel 2098  βˆ€wral 3053  Vcvv 3466   βŠ† wss 3941   β†Ύ cres 5669   ∘ ccom 5671   Fn wfn 6529  βŸΆwf 6530  β€“ontoβ†’wfo 6532  β€˜cfv 6534  1st c1st 7967  Basecbs 17145  domacdoma 17974  Arrowcarw 17976
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-rep 5276  ax-sep 5290  ax-nul 5297  ax-pow 5354  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-reu 3369  df-rab 3425  df-v 3468  df-sbc 3771  df-csb 3887  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-iun 4990  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-fv 6542  df-ov 7405  df-1st 7969  df-2nd 7970  df-doma 17978  df-coda 17979  df-homa 17980  df-arw 17981
This theorem is referenced by: (None)
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