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Theorem coapm 18116
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 2737 . . . . . 6 (comp‘𝐶) = (comp‘𝐶)
41, 2, 3coafval 18109 . . . . 5 · = (𝑔𝐴, 𝑓 ∈ {𝐴 ∣ (coda) = (doma𝑔)} ↦ ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩(comp‘𝐶)(coda𝑔))(2nd𝑓))⟩)
54mpofun 7557 . . . 4 Fun ·
6 funfn 6596 . . . 4 (Fun ·· Fn dom · )
75, 6mpbi 230 . . 3 · Fn dom ·
81, 2dmcoass 18111 . . . . . . . . 9 dom · ⊆ (𝐴 × 𝐴)
98sseli 3979 . . . . . . . 8 (𝑧 ∈ dom ·𝑧 ∈ (𝐴 × 𝐴))
10 1st2nd2 8053 . . . . . . . 8 (𝑧 ∈ (𝐴 × 𝐴) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
119, 10syl 17 . . . . . . 7 (𝑧 ∈ dom ·𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
1211fveq2d 6910 . . . . . 6 (𝑧 ∈ dom · → ( ·𝑧) = ( · ‘⟨(1st𝑧), (2nd𝑧)⟩))
13 df-ov 7434 . . . . . 6 ((1st𝑧) · (2nd𝑧)) = ( · ‘⟨(1st𝑧), (2nd𝑧)⟩)
1412, 13eqtr4di 2795 . . . . 5 (𝑧 ∈ dom · → ( ·𝑧) = ((1st𝑧) · (2nd𝑧)))
15 eqid 2737 . . . . . . 7 (Homa𝐶) = (Homa𝐶)
162, 15homarw 18091 . . . . . 6 ((doma‘(2nd𝑧))(Homa𝐶)(coda‘(1st𝑧))) ⊆ 𝐴
17 id 22 . . . . . . . . . . . . 13 (𝑧 ∈ dom ·𝑧 ∈ dom · )
1811, 17eqeltrrd 2842 . . . . . . . . . . . 12 (𝑧 ∈ dom · → ⟨(1st𝑧), (2nd𝑧)⟩ ∈ dom · )
19 df-br 5144 . . . . . . . . . . . 12 ((1st𝑧)dom · (2nd𝑧) ↔ ⟨(1st𝑧), (2nd𝑧)⟩ ∈ dom · )
2018, 19sylibr 234 . . . . . . . . . . 11 (𝑧 ∈ dom · → (1st𝑧)dom · (2nd𝑧))
211, 2eldmcoa 18110 . . . . . . . . . . 11 ((1st𝑧)dom · (2nd𝑧) ↔ ((2nd𝑧) ∈ 𝐴 ∧ (1st𝑧) ∈ 𝐴 ∧ (coda‘(2nd𝑧)) = (doma‘(1st𝑧))))
2220, 21sylib 218 . . . . . . . . . 10 (𝑧 ∈ dom · → ((2nd𝑧) ∈ 𝐴 ∧ (1st𝑧) ∈ 𝐴 ∧ (coda‘(2nd𝑧)) = (doma‘(1st𝑧))))
2322simp1d 1143 . . . . . . . . 9 (𝑧 ∈ dom · → (2nd𝑧) ∈ 𝐴)
242, 15arwhoma 18090 . . . . . . . . 9 ((2nd𝑧) ∈ 𝐴 → (2nd𝑧) ∈ ((doma‘(2nd𝑧))(Homa𝐶)(coda‘(2nd𝑧))))
2523, 24syl 17 . . . . . . . 8 (𝑧 ∈ dom · → (2nd𝑧) ∈ ((doma‘(2nd𝑧))(Homa𝐶)(coda‘(2nd𝑧))))
2622simp3d 1145 . . . . . . . . 9 (𝑧 ∈ dom · → (coda‘(2nd𝑧)) = (doma‘(1st𝑧)))
2726oveq2d 7447 . . . . . . . 8 (𝑧 ∈ dom · → ((doma‘(2nd𝑧))(Homa𝐶)(coda‘(2nd𝑧))) = ((doma‘(2nd𝑧))(Homa𝐶)(doma‘(1st𝑧))))
2825, 27eleqtrd 2843 . . . . . . 7 (𝑧 ∈ dom · → (2nd𝑧) ∈ ((doma‘(2nd𝑧))(Homa𝐶)(doma‘(1st𝑧))))
2922simp2d 1144 . . . . . . . 8 (𝑧 ∈ dom · → (1st𝑧) ∈ 𝐴)
302, 15arwhoma 18090 . . . . . . . 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 18115 . . . . . 6 (𝑧 ∈ dom · → ((1st𝑧) · (2nd𝑧)) ∈ ((doma‘(2nd𝑧))(Homa𝐶)(coda‘(1st𝑧))))
3316, 32sselid 3981 . . . . 5 (𝑧 ∈ dom · → ((1st𝑧) · (2nd𝑧)) ∈ 𝐴)
3414, 33eqeltrd 2841 . . . 4 (𝑧 ∈ dom · → ( ·𝑧) ∈ 𝐴)
3534rgen 3063 . . 3 𝑧 ∈ dom · ( ·𝑧) ∈ 𝐴
36 ffnfv 7139 . . 3 ( · :dom ·𝐴 ↔ ( · Fn dom · ∧ ∀𝑧 ∈ dom · ( ·𝑧) ∈ 𝐴))
377, 35, 36mpbir2an 711 . 2 · :dom ·𝐴
382fvexi 6920 . . 3 𝐴 ∈ V
3938, 38xpex 7773 . . 3 (𝐴 × 𝐴) ∈ V
4038, 39elpm2 8914 . 2 ( · ∈ (𝐴pm (𝐴 × 𝐴)) ↔ ( · :dom ·𝐴 ∧ dom · ⊆ (𝐴 × 𝐴)))
4137, 8, 40mpbir2an 711 1 · ∈ (𝐴pm (𝐴 × 𝐴))
Colors of variables: wff setvar class
Syntax hints:  w3a 1087   = wceq 1540  wcel 2108  wral 3061  {crab 3436  wss 3951  cop 4632  cotp 4634   class class class wbr 5143   × cxp 5683  dom cdm 5685  Fun wfun 6555   Fn wfn 6556  wf 6557  cfv 6561  (class class class)co 7431  1st c1st 8012  2nd c2nd 8013  pm cpm 8867  compcco 17309  domacdoma 18065  codaccoda 18066  Arrowcarw 18067  Homachoma 18068  compaccoa 18099
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-ot 4635  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8014  df-2nd 8015  df-pm 8869  df-cat 17711  df-doma 18069  df-coda 18070  df-homa 18071  df-arw 18072  df-coa 18101
This theorem is referenced by: (None)
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