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Theorem dmaf 17940
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 7942 . . . . . 6 1st :V–ontoβ†’V
2 fofn 6759 . . . . . 6 (1st :V–ontoβ†’V β†’ 1st Fn V)
31, 2ax-mp 5 . . . . 5 1st Fn V
4 fof 6757 . . . . . 6 (1st :V–ontoβ†’V β†’ 1st :V⟢V)
51, 4ax-mp 5 . . . . 5 1st :V⟢V
6 fnfco 6708 . . . . 5 ((1st Fn V ∧ 1st :V⟢V) β†’ (1st ∘ 1st ) Fn V)
73, 5, 6mp2an 691 . . . 4 (1st ∘ 1st ) Fn V
8 df-doma 17915 . . . . 5 doma = (1st ∘ 1st )
98fneq1i 6600 . . . 4 (doma Fn V ↔ (1st ∘ 1st ) Fn V)
107, 9mpbir 230 . . 3 doma Fn V
11 ssv 3969 . . 3 𝐴 βŠ† V
12 fnssres 6625 . . 3 ((doma Fn V ∧ 𝐴 βŠ† V) β†’ (doma β†Ύ 𝐴) Fn 𝐴)
1310, 11, 12mp2an 691 . 2 (doma β†Ύ 𝐴) Fn 𝐴
14 fvres 6862 . . . 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 17938 . . . 4 (π‘₯ ∈ 𝐴 β†’ (domaβ€˜π‘₯) ∈ 𝐡)
1814, 17eqeltrd 2834 . . 3 (π‘₯ ∈ 𝐴 β†’ ((doma β†Ύ 𝐴)β€˜π‘₯) ∈ 𝐡)
1918rgen 3063 . 2 βˆ€π‘₯ ∈ 𝐴 ((doma β†Ύ 𝐴)β€˜π‘₯) ∈ 𝐡
20 ffnfv 7067 . 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 710 1 (doma β†Ύ 𝐴):𝐴⟢𝐡
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  Vcvv 3444   βŠ† wss 3911   β†Ύ cres 5636   ∘ ccom 5638   Fn wfn 6492  βŸΆwf 6493  β€“ontoβ†’wfo 6495  β€˜cfv 6497  1st c1st 7920  Basecbs 17088  domacdoma 17911  Arrowcarw 17913
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-1st 7922  df-2nd 7923  df-doma 17915  df-coda 17916  df-homa 17917  df-arw 17918
This theorem is referenced by: (None)
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