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Theorem dmaf 17301
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 7701 . . . . . 6 1st :V–onto→V
2 fofn 6585 . . . . . 6 (1st :V–onto→V → 1st Fn V)
31, 2ax-mp 5 . . . . 5 1st Fn V
4 fof 6583 . . . . . 6 (1st :V–onto→V → 1st :V⟶V)
51, 4ax-mp 5 . . . . 5 1st :V⟶V
6 fnfco 6536 . . . . 5 ((1st Fn V ∧ 1st :V⟶V) → (1st ∘ 1st ) Fn V)
73, 5, 6mp2an 690 . . . 4 (1st ∘ 1st ) Fn V
8 df-doma 17276 . . . . 5 doma = (1st ∘ 1st )
98fneq1i 6443 . . . 4 (doma Fn V ↔ (1st ∘ 1st ) Fn V)
107, 9mpbir 233 . . 3 doma Fn V
11 ssv 3989 . . 3 𝐴 ⊆ V
12 fnssres 6463 . . 3 ((doma Fn V ∧ 𝐴 ⊆ V) → (doma𝐴) Fn 𝐴)
1310, 11, 12mp2an 690 . 2 (doma𝐴) Fn 𝐴
14 fvres 6682 . . . 4 (𝑥𝐴 → ((doma𝐴)‘𝑥) = (doma𝑥))
15 arwrcl.a . . . . 5 𝐴 = (Arrow‘𝐶)
16 arwdm.b . . . . 5 𝐵 = (Base‘𝐶)
1715, 16arwdm 17299 . . . 4 (𝑥𝐴 → (doma𝑥) ∈ 𝐵)
1814, 17eqeltrd 2911 . . 3 (𝑥𝐴 → ((doma𝐴)‘𝑥) ∈ 𝐵)
1918rgen 3146 . 2 𝑥𝐴 ((doma𝐴)‘𝑥) ∈ 𝐵
20 ffnfv 6875 . 2 ((doma𝐴):𝐴𝐵 ↔ ((doma𝐴) Fn 𝐴 ∧ ∀𝑥𝐴 ((doma𝐴)‘𝑥) ∈ 𝐵))
2113, 19, 20mpbir2an 709 1 (doma𝐴):𝐴𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1530  wcel 2107  wral 3136  Vcvv 3493  wss 3934  cres 5550  ccom 5552   Fn wfn 6343  wf 6344  ontowfo 6346  cfv 6348  1st c1st 7679  Basecbs 16475  domacdoma 17272  Arrowcarw 17274
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7151  df-1st 7681  df-2nd 7682  df-doma 17276  df-coda 17277  df-homa 17278  df-arw 17279
This theorem is referenced by: (None)
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