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Theorem dmaf 18031
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 8007 . . . . . 6 1st :V–ontoβ†’V
2 fofn 6807 . . . . . 6 (1st :V–ontoβ†’V β†’ 1st Fn V)
31, 2ax-mp 5 . . . . 5 1st Fn V
4 fof 6805 . . . . . 6 (1st :V–ontoβ†’V β†’ 1st :V⟢V)
51, 4ax-mp 5 . . . . 5 1st :V⟢V
6 fnfco 6756 . . . . 5 ((1st Fn V ∧ 1st :V⟢V) β†’ (1st ∘ 1st ) Fn V)
73, 5, 6mp2an 691 . . . 4 (1st ∘ 1st ) Fn V
8 df-doma 18006 . . . . 5 doma = (1st ∘ 1st )
98fneq1i 6645 . . . 4 (doma Fn V ↔ (1st ∘ 1st ) Fn V)
107, 9mpbir 230 . . 3 doma Fn V
11 ssv 4002 . . 3 𝐴 βŠ† V
12 fnssres 6672 . . 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 6910 . . . 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 18029 . . . 4 (π‘₯ ∈ 𝐴 β†’ (domaβ€˜π‘₯) ∈ 𝐡)
1814, 17eqeltrd 2829 . . 3 (π‘₯ ∈ 𝐴 β†’ ((doma β†Ύ 𝐴)β€˜π‘₯) ∈ 𝐡)
1918rgen 3059 . 2 βˆ€π‘₯ ∈ 𝐴 ((doma β†Ύ 𝐴)β€˜π‘₯) ∈ 𝐡
20 ffnfv 7123 . 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 1534   ∈ wcel 2099  βˆ€wral 3057  Vcvv 3470   βŠ† wss 3945   β†Ύ cres 5674   ∘ ccom 5676   Fn wfn 6537  βŸΆwf 6538  β€“ontoβ†’wfo 6540  β€˜cfv 6542  1st c1st 7985  Basecbs 17173  domacdoma 18002  Arrowcarw 18004
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-rep 5279  ax-sep 5293  ax-nul 5300  ax-pow 5359  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-reu 3373  df-rab 3429  df-v 3472  df-sbc 3776  df-csb 3891  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-mpt 5226  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-ov 7417  df-1st 7987  df-2nd 7988  df-doma 18006  df-coda 18007  df-homa 18008  df-arw 18009
This theorem is referenced by: (None)
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