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Theorem cdaf 16921
 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 7355 . . . . . 6 2nd :V–onto→V
2 fofn 6279 . . . . . 6 (2nd :V–onto→V → 2nd Fn V)
31, 2ax-mp 5 . . . . 5 2nd Fn V
4 fo1st 7354 . . . . . 6 1st :V–onto→V
5 fof 6277 . . . . . 6 (1st :V–onto→V → 1st :V⟶V)
64, 5ax-mp 5 . . . . 5 1st :V⟶V
7 fnfco 6230 . . . . 5 ((2nd Fn V ∧ 1st :V⟶V) → (2nd ∘ 1st ) Fn V)
83, 6, 7mp2an 710 . . . 4 (2nd ∘ 1st ) Fn V
9 df-coda 16896 . . . . 5 coda = (2nd ∘ 1st )
109fneq1i 6146 . . . 4 (coda Fn V ↔ (2nd ∘ 1st ) Fn V)
118, 10mpbir 221 . . 3 coda Fn V
12 ssv 3766 . . 3 𝐴 ⊆ V
13 fnssres 6165 . . 3 ((coda Fn V ∧ 𝐴 ⊆ V) → (coda𝐴) Fn 𝐴)
1411, 12, 13mp2an 710 . 2 (coda𝐴) Fn 𝐴
15 fvres 6369 . . . 4 (𝑥𝐴 → ((coda𝐴)‘𝑥) = (coda𝑥))
16 arwrcl.a . . . . 5 𝐴 = (Arrow‘𝐶)
17 arwdm.b . . . . 5 𝐵 = (Base‘𝐶)
1816, 17arwcd 16919 . . . 4 (𝑥𝐴 → (coda𝑥) ∈ 𝐵)
1915, 18eqeltrd 2839 . . 3 (𝑥𝐴 → ((coda𝐴)‘𝑥) ∈ 𝐵)
2019rgen 3060 . 2 𝑥𝐴 ((coda𝐴)‘𝑥) ∈ 𝐵
21 ffnfv 6552 . 2 ((coda𝐴):𝐴𝐵 ↔ ((coda𝐴) Fn 𝐴 ∧ ∀𝑥𝐴 ((coda𝐴)‘𝑥) ∈ 𝐵))
2214, 20, 21mpbir2an 993 1 (coda𝐴):𝐴𝐵
 Colors of variables: wff setvar class Syntax hints:   = wceq 1632   ∈ wcel 2139  ∀wral 3050  Vcvv 3340   ⊆ wss 3715   ↾ cres 5268   ∘ ccom 5270   Fn wfn 6044  ⟶wf 6045  –onto→wfo 6047  ‘cfv 6049  1st c1st 7332  2nd c2nd 7333  Basecbs 16079  codaccoda 16892  Arrowcarw 16893 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-1st 7334  df-2nd 7335  df-doma 16895  df-coda 16896  df-homa 16897  df-arw 16898 This theorem is referenced by: (None)
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