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Theorem cdaf 17302
Description: The codomain 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
cdaf (coda𝐴):𝐴𝐵

Proof of Theorem cdaf
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 fo2nd 7704 . . . . . 6 2nd :V–onto→V
2 fofn 6588 . . . . . 6 (2nd :V–onto→V → 2nd Fn V)
31, 2ax-mp 5 . . . . 5 2nd Fn V
4 fo1st 7703 . . . . . 6 1st :V–onto→V
5 fof 6586 . . . . . 6 (1st :V–onto→V → 1st :V⟶V)
64, 5ax-mp 5 . . . . 5 1st :V⟶V
7 fnfco 6539 . . . . 5 ((2nd Fn V ∧ 1st :V⟶V) → (2nd ∘ 1st ) Fn V)
83, 6, 7mp2an 688 . . . 4 (2nd ∘ 1st ) Fn V
9 df-coda 17277 . . . . 5 coda = (2nd ∘ 1st )
109fneq1i 6446 . . . 4 (coda Fn V ↔ (2nd ∘ 1st ) Fn V)
118, 10mpbir 232 . . 3 coda Fn V
12 ssv 3994 . . 3 𝐴 ⊆ V
13 fnssres 6466 . . 3 ((coda Fn V ∧ 𝐴 ⊆ V) → (coda𝐴) Fn 𝐴)
1411, 12, 13mp2an 688 . 2 (coda𝐴) Fn 𝐴
15 fvres 6685 . . . 4 (𝑥𝐴 → ((coda𝐴)‘𝑥) = (coda𝑥))
16 arwrcl.a . . . . 5 𝐴 = (Arrow‘𝐶)
17 arwdm.b . . . . 5 𝐵 = (Base‘𝐶)
1816, 17arwcd 17300 . . . 4 (𝑥𝐴 → (coda𝑥) ∈ 𝐵)
1915, 18eqeltrd 2917 . . 3 (𝑥𝐴 → ((coda𝐴)‘𝑥) ∈ 𝐵)
2019rgen 3152 . 2 𝑥𝐴 ((coda𝐴)‘𝑥) ∈ 𝐵
21 ffnfv 6877 . 2 ((coda𝐴):𝐴𝐵 ↔ ((coda𝐴) Fn 𝐴 ∧ ∀𝑥𝐴 ((coda𝐴)‘𝑥) ∈ 𝐵))
2214, 20, 21mpbir2an 707 1 (coda𝐴):𝐴𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1530  wcel 2106  wral 3142  Vcvv 3499  wss 3939  cres 5555  ccom 5557   Fn wfn 6346  wf 6347  ontowfo 6349  cfv 6351  1st c1st 7681  2nd c2nd 7682  Basecbs 16475  codaccoda 17273  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 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-rep 5186  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-pw 4543  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-ov 7154  df-1st 7683  df-2nd 7684  df-doma 17276  df-coda 17277  df-homa 17278  df-arw 17279
This theorem is referenced by: (None)
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