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Theorem coapm 16544
 Description: Composition of arrows is a partial binary operation on arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
coapm.o · = (compa𝐶)
coapm.a 𝐴 = (Arrow‘𝐶)
Assertion
Ref Expression
coapm · ∈ (𝐴pm (𝐴 × 𝐴))

Proof of Theorem coapm
Dummy variables 𝑓 𝑔 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 coapm.o . . . . . 6 · = (compa𝐶)
2 coapm.a . . . . . 6 𝐴 = (Arrow‘𝐶)
3 eqid 2610 . . . . . 6 (comp‘𝐶) = (comp‘𝐶)
41, 2, 3coafval 16537 . . . . 5 · = (𝑔𝐴, 𝑓 ∈ {𝐴 ∣ (coda) = (doma𝑔)} ↦ ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩(comp‘𝐶)(coda𝑔))(2nd𝑓))⟩)
54mpt2fun 6660 . . . 4 Fun ·
6 funfn 5833 . . . 4 (Fun ·· Fn dom · )
75, 6mpbi 219 . . 3 · Fn dom ·
81, 2dmcoass 16539 . . . . . . . . 9 dom · ⊆ (𝐴 × 𝐴)
98sseli 3564 . . . . . . . 8 (𝑧 ∈ dom ·𝑧 ∈ (𝐴 × 𝐴))
10 1st2nd2 7096 . . . . . . . 8 (𝑧 ∈ (𝐴 × 𝐴) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
119, 10syl 17 . . . . . . 7 (𝑧 ∈ dom ·𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
1211fveq2d 6107 . . . . . 6 (𝑧 ∈ dom · → ( ·𝑧) = ( · ‘⟨(1st𝑧), (2nd𝑧)⟩))
13 df-ov 6552 . . . . . 6 ((1st𝑧) · (2nd𝑧)) = ( · ‘⟨(1st𝑧), (2nd𝑧)⟩)
1412, 13syl6eqr 2662 . . . . 5 (𝑧 ∈ dom · → ( ·𝑧) = ((1st𝑧) · (2nd𝑧)))
15 eqid 2610 . . . . . . 7 (Homa𝐶) = (Homa𝐶)
162, 15homarw 16519 . . . . . 6 ((doma‘(2nd𝑧))(Homa𝐶)(coda‘(1st𝑧))) ⊆ 𝐴
17 id 22 . . . . . . . . . . . . 13 (𝑧 ∈ dom ·𝑧 ∈ dom · )
1811, 17eqeltrrd 2689 . . . . . . . . . . . 12 (𝑧 ∈ dom · → ⟨(1st𝑧), (2nd𝑧)⟩ ∈ dom · )
19 df-br 4584 . . . . . . . . . . . 12 ((1st𝑧)dom · (2nd𝑧) ↔ ⟨(1st𝑧), (2nd𝑧)⟩ ∈ dom · )
2018, 19sylibr 223 . . . . . . . . . . 11 (𝑧 ∈ dom · → (1st𝑧)dom · (2nd𝑧))
211, 2eldmcoa 16538 . . . . . . . . . . 11 ((1st𝑧)dom · (2nd𝑧) ↔ ((2nd𝑧) ∈ 𝐴 ∧ (1st𝑧) ∈ 𝐴 ∧ (coda‘(2nd𝑧)) = (doma‘(1st𝑧))))
2220, 21sylib 207 . . . . . . . . . 10 (𝑧 ∈ dom · → ((2nd𝑧) ∈ 𝐴 ∧ (1st𝑧) ∈ 𝐴 ∧ (coda‘(2nd𝑧)) = (doma‘(1st𝑧))))
2322simp1d 1066 . . . . . . . . 9 (𝑧 ∈ dom · → (2nd𝑧) ∈ 𝐴)
242, 15arwhoma 16518 . . . . . . . . 9 ((2nd𝑧) ∈ 𝐴 → (2nd𝑧) ∈ ((doma‘(2nd𝑧))(Homa𝐶)(coda‘(2nd𝑧))))
2523, 24syl 17 . . . . . . . 8 (𝑧 ∈ dom · → (2nd𝑧) ∈ ((doma‘(2nd𝑧))(Homa𝐶)(coda‘(2nd𝑧))))
2622simp3d 1068 . . . . . . . . 9 (𝑧 ∈ dom · → (coda‘(2nd𝑧)) = (doma‘(1st𝑧)))
2726oveq2d 6565 . . . . . . . 8 (𝑧 ∈ dom · → ((doma‘(2nd𝑧))(Homa𝐶)(coda‘(2nd𝑧))) = ((doma‘(2nd𝑧))(Homa𝐶)(doma‘(1st𝑧))))
2825, 27eleqtrd 2690 . . . . . . 7 (𝑧 ∈ dom · → (2nd𝑧) ∈ ((doma‘(2nd𝑧))(Homa𝐶)(doma‘(1st𝑧))))
2922simp2d 1067 . . . . . . . 8 (𝑧 ∈ dom · → (1st𝑧) ∈ 𝐴)
302, 15arwhoma 16518 . . . . . . . 8 ((1st𝑧) ∈ 𝐴 → (1st𝑧) ∈ ((doma‘(1st𝑧))(Homa𝐶)(coda‘(1st𝑧))))
3129, 30syl 17 . . . . . . 7 (𝑧 ∈ dom · → (1st𝑧) ∈ ((doma‘(1st𝑧))(Homa𝐶)(coda‘(1st𝑧))))
321, 15, 28, 31coahom 16543 . . . . . 6 (𝑧 ∈ dom · → ((1st𝑧) · (2nd𝑧)) ∈ ((doma‘(2nd𝑧))(Homa𝐶)(coda‘(1st𝑧))))
3316, 32sseldi 3566 . . . . 5 (𝑧 ∈ dom · → ((1st𝑧) · (2nd𝑧)) ∈ 𝐴)
3414, 33eqeltrd 2688 . . . 4 (𝑧 ∈ dom · → ( ·𝑧) ∈ 𝐴)
3534rgen 2906 . . 3 𝑧 ∈ dom · ( ·𝑧) ∈ 𝐴
36 ffnfv 6295 . . 3 ( · :dom ·𝐴 ↔ ( · Fn dom · ∧ ∀𝑧 ∈ dom · ( ·𝑧) ∈ 𝐴))
377, 35, 36mpbir2an 957 . 2 · :dom ·𝐴
38 fvex 6113 . . . 4 (Arrow‘𝐶) ∈ V
392, 38eqeltri 2684 . . 3 𝐴 ∈ V
4039, 39xpex 6860 . . 3 (𝐴 × 𝐴) ∈ V
4139, 40elpm2 7775 . 2 ( · ∈ (𝐴pm (𝐴 × 𝐴)) ↔ ( · :dom ·𝐴 ∧ dom · ⊆ (𝐴 × 𝐴)))
4237, 8, 41mpbir2an 957 1 · ∈ (𝐴pm (𝐴 × 𝐴))
 Colors of variables: wff setvar class Syntax hints:   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∀wral 2896  {crab 2900  Vcvv 3173   ⊆ wss 3540  ⟨cop 4131  ⟨cotp 4133   class class class wbr 4583   × cxp 5036  dom cdm 5038  Fun wfun 5798   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058   ↑pm cpm 7745  compcco 15780  domacdoma 16493  codaccoda 16494  Arrowcarw 16495  Homachoma 16496  compaccoa 16527 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-ot 4134  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-pm 7747  df-cat 16152  df-doma 16497  df-coda 16498  df-homa 16499  df-arw 16500  df-coa 16529 This theorem is referenced by: (None)
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