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Theorem coapm 17326
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 2826 . . . . . 6 (comp‘𝐶) = (comp‘𝐶)
41, 2, 3coafval 17319 . . . . 5 · = (𝑔𝐴, 𝑓 ∈ {𝐴 ∣ (coda) = (doma𝑔)} ↦ ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩(comp‘𝐶)(coda𝑔))(2nd𝑓))⟩)
54mpofun 7270 . . . 4 Fun ·
6 funfn 6384 . . . 4 (Fun ·· Fn dom · )
75, 6mpbi 231 . . 3 · Fn dom ·
81, 2dmcoass 17321 . . . . . . . . 9 dom · ⊆ (𝐴 × 𝐴)
98sseli 3967 . . . . . . . 8 (𝑧 ∈ dom ·𝑧 ∈ (𝐴 × 𝐴))
10 1st2nd2 7724 . . . . . . . 8 (𝑧 ∈ (𝐴 × 𝐴) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
119, 10syl 17 . . . . . . 7 (𝑧 ∈ dom ·𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
1211fveq2d 6673 . . . . . 6 (𝑧 ∈ dom · → ( ·𝑧) = ( · ‘⟨(1st𝑧), (2nd𝑧)⟩))
13 df-ov 7153 . . . . . 6 ((1st𝑧) · (2nd𝑧)) = ( · ‘⟨(1st𝑧), (2nd𝑧)⟩)
1412, 13syl6eqr 2879 . . . . 5 (𝑧 ∈ dom · → ( ·𝑧) = ((1st𝑧) · (2nd𝑧)))
15 eqid 2826 . . . . . . 7 (Homa𝐶) = (Homa𝐶)
162, 15homarw 17301 . . . . . 6 ((doma‘(2nd𝑧))(Homa𝐶)(coda‘(1st𝑧))) ⊆ 𝐴
17 id 22 . . . . . . . . . . . . 13 (𝑧 ∈ dom ·𝑧 ∈ dom · )
1811, 17eqeltrrd 2919 . . . . . . . . . . . 12 (𝑧 ∈ dom · → ⟨(1st𝑧), (2nd𝑧)⟩ ∈ dom · )
19 df-br 5064 . . . . . . . . . . . 12 ((1st𝑧)dom · (2nd𝑧) ↔ ⟨(1st𝑧), (2nd𝑧)⟩ ∈ dom · )
2018, 19sylibr 235 . . . . . . . . . . 11 (𝑧 ∈ dom · → (1st𝑧)dom · (2nd𝑧))
211, 2eldmcoa 17320 . . . . . . . . . . 11 ((1st𝑧)dom · (2nd𝑧) ↔ ((2nd𝑧) ∈ 𝐴 ∧ (1st𝑧) ∈ 𝐴 ∧ (coda‘(2nd𝑧)) = (doma‘(1st𝑧))))
2220, 21sylib 219 . . . . . . . . . 10 (𝑧 ∈ dom · → ((2nd𝑧) ∈ 𝐴 ∧ (1st𝑧) ∈ 𝐴 ∧ (coda‘(2nd𝑧)) = (doma‘(1st𝑧))))
2322simp1d 1136 . . . . . . . . 9 (𝑧 ∈ dom · → (2nd𝑧) ∈ 𝐴)
242, 15arwhoma 17300 . . . . . . . . 9 ((2nd𝑧) ∈ 𝐴 → (2nd𝑧) ∈ ((doma‘(2nd𝑧))(Homa𝐶)(coda‘(2nd𝑧))))
2523, 24syl 17 . . . . . . . 8 (𝑧 ∈ dom · → (2nd𝑧) ∈ ((doma‘(2nd𝑧))(Homa𝐶)(coda‘(2nd𝑧))))
2622simp3d 1138 . . . . . . . . 9 (𝑧 ∈ dom · → (coda‘(2nd𝑧)) = (doma‘(1st𝑧)))
2726oveq2d 7166 . . . . . . . 8 (𝑧 ∈ dom · → ((doma‘(2nd𝑧))(Homa𝐶)(coda‘(2nd𝑧))) = ((doma‘(2nd𝑧))(Homa𝐶)(doma‘(1st𝑧))))
2825, 27eleqtrd 2920 . . . . . . 7 (𝑧 ∈ dom · → (2nd𝑧) ∈ ((doma‘(2nd𝑧))(Homa𝐶)(doma‘(1st𝑧))))
2922simp2d 1137 . . . . . . . 8 (𝑧 ∈ dom · → (1st𝑧) ∈ 𝐴)
302, 15arwhoma 17300 . . . . . . . 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 17325 . . . . . 6 (𝑧 ∈ dom · → ((1st𝑧) · (2nd𝑧)) ∈ ((doma‘(2nd𝑧))(Homa𝐶)(coda‘(1st𝑧))))
3316, 32sseldi 3969 . . . . 5 (𝑧 ∈ dom · → ((1st𝑧) · (2nd𝑧)) ∈ 𝐴)
3414, 33eqeltrd 2918 . . . 4 (𝑧 ∈ dom · → ( ·𝑧) ∈ 𝐴)
3534rgen 3153 . . 3 𝑧 ∈ dom · ( ·𝑧) ∈ 𝐴
36 ffnfv 6880 . . 3 ( · :dom ·𝐴 ↔ ( · Fn dom · ∧ ∀𝑧 ∈ dom · ( ·𝑧) ∈ 𝐴))
377, 35, 36mpbir2an 707 . 2 · :dom ·𝐴
382fvexi 6683 . . 3 𝐴 ∈ V
3938, 38xpex 7469 . . 3 (𝐴 × 𝐴) ∈ V
4038, 39elpm2 8433 . 2 ( · ∈ (𝐴pm (𝐴 × 𝐴)) ↔ ( · :dom ·𝐴 ∧ dom · ⊆ (𝐴 × 𝐴)))
4137, 8, 40mpbir2an 707 1 · ∈ (𝐴pm (𝐴 × 𝐴))
Colors of variables: wff setvar class
Syntax hints:  w3a 1081   = wceq 1530  wcel 2107  wral 3143  {crab 3147  wss 3940  cop 4570  cotp 4572   class class class wbr 5063   × cxp 5552  dom cdm 5554  Fun wfun 6348   Fn wfn 6349  wf 6350  cfv 6354  (class class class)co 7150  1st c1st 7683  2nd c2nd 7684  pm cpm 8402  compcco 16572  domacdoma 17275  codaccoda 17276  Arrowcarw 17277  Homachoma 17278  compaccoa 17309
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 2798  ax-rep 5187  ax-sep 5200  ax-nul 5207  ax-pow 5263  ax-pr 5326  ax-un 7455
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 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ne 3022  df-ral 3148  df-rex 3149  df-reu 3150  df-rab 3152  df-v 3502  df-sbc 3777  df-csb 3888  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-pw 4544  df-sn 4565  df-pr 4567  df-op 4571  df-ot 4573  df-uni 4838  df-iun 4919  df-br 5064  df-opab 5126  df-mpt 5144  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-iota 6313  df-fun 6356  df-fn 6357  df-f 6358  df-f1 6359  df-fo 6360  df-f1o 6361  df-fv 6362  df-ov 7153  df-oprab 7154  df-mpo 7155  df-1st 7685  df-2nd 7686  df-pm 8404  df-cat 16934  df-doma 17279  df-coda 17280  df-homa 17281  df-arw 17282  df-coa 17311
This theorem is referenced by: (None)
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