MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  coapm Structured version   Visualization version   GIF version

Theorem coapm 17436
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 2738 . . . . . 6 (comp‘𝐶) = (comp‘𝐶)
41, 2, 3coafval 17429 . . . . 5 · = (𝑔𝐴, 𝑓 ∈ {𝐴 ∣ (coda) = (doma𝑔)} ↦ ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩(comp‘𝐶)(coda𝑔))(2nd𝑓))⟩)
54mpofun 7284 . . . 4 Fun ·
6 funfn 6363 . . . 4 (Fun ·· Fn dom · )
75, 6mpbi 233 . . 3 · Fn dom ·
81, 2dmcoass 17431 . . . . . . . . 9 dom · ⊆ (𝐴 × 𝐴)
98sseli 3871 . . . . . . . 8 (𝑧 ∈ dom ·𝑧 ∈ (𝐴 × 𝐴))
10 1st2nd2 7746 . . . . . . . 8 (𝑧 ∈ (𝐴 × 𝐴) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
119, 10syl 17 . . . . . . 7 (𝑧 ∈ dom ·𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
1211fveq2d 6672 . . . . . 6 (𝑧 ∈ dom · → ( ·𝑧) = ( · ‘⟨(1st𝑧), (2nd𝑧)⟩))
13 df-ov 7167 . . . . . 6 ((1st𝑧) · (2nd𝑧)) = ( · ‘⟨(1st𝑧), (2nd𝑧)⟩)
1412, 13eqtr4di 2791 . . . . 5 (𝑧 ∈ dom · → ( ·𝑧) = ((1st𝑧) · (2nd𝑧)))
15 eqid 2738 . . . . . . 7 (Homa𝐶) = (Homa𝐶)
162, 15homarw 17411 . . . . . 6 ((doma‘(2nd𝑧))(Homa𝐶)(coda‘(1st𝑧))) ⊆ 𝐴
17 id 22 . . . . . . . . . . . . 13 (𝑧 ∈ dom ·𝑧 ∈ dom · )
1811, 17eqeltrrd 2834 . . . . . . . . . . . 12 (𝑧 ∈ dom · → ⟨(1st𝑧), (2nd𝑧)⟩ ∈ dom · )
19 df-br 5028 . . . . . . . . . . . 12 ((1st𝑧)dom · (2nd𝑧) ↔ ⟨(1st𝑧), (2nd𝑧)⟩ ∈ dom · )
2018, 19sylibr 237 . . . . . . . . . . 11 (𝑧 ∈ dom · → (1st𝑧)dom · (2nd𝑧))
211, 2eldmcoa 17430 . . . . . . . . . . 11 ((1st𝑧)dom · (2nd𝑧) ↔ ((2nd𝑧) ∈ 𝐴 ∧ (1st𝑧) ∈ 𝐴 ∧ (coda‘(2nd𝑧)) = (doma‘(1st𝑧))))
2220, 21sylib 221 . . . . . . . . . 10 (𝑧 ∈ dom · → ((2nd𝑧) ∈ 𝐴 ∧ (1st𝑧) ∈ 𝐴 ∧ (coda‘(2nd𝑧)) = (doma‘(1st𝑧))))
2322simp1d 1143 . . . . . . . . 9 (𝑧 ∈ dom · → (2nd𝑧) ∈ 𝐴)
242, 15arwhoma 17410 . . . . . . . . 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 7180 . . . . . . . 8 (𝑧 ∈ dom · → ((doma‘(2nd𝑧))(Homa𝐶)(coda‘(2nd𝑧))) = ((doma‘(2nd𝑧))(Homa𝐶)(doma‘(1st𝑧))))
2825, 27eleqtrd 2835 . . . . . . 7 (𝑧 ∈ dom · → (2nd𝑧) ∈ ((doma‘(2nd𝑧))(Homa𝐶)(doma‘(1st𝑧))))
2922simp2d 1144 . . . . . . . 8 (𝑧 ∈ dom · → (1st𝑧) ∈ 𝐴)
302, 15arwhoma 17410 . . . . . . . 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 17435 . . . . . 6 (𝑧 ∈ dom · → ((1st𝑧) · (2nd𝑧)) ∈ ((doma‘(2nd𝑧))(Homa𝐶)(coda‘(1st𝑧))))
3316, 32sseldi 3873 . . . . 5 (𝑧 ∈ dom · → ((1st𝑧) · (2nd𝑧)) ∈ 𝐴)
3414, 33eqeltrd 2833 . . . 4 (𝑧 ∈ dom · → ( ·𝑧) ∈ 𝐴)
3534rgen 3063 . . 3 𝑧 ∈ dom · ( ·𝑧) ∈ 𝐴
36 ffnfv 6886 . . 3 ( · :dom ·𝐴 ↔ ( · Fn dom · ∧ ∀𝑧 ∈ dom · ( ·𝑧) ∈ 𝐴))
377, 35, 36mpbir2an 711 . 2 · :dom ·𝐴
382fvexi 6682 . . 3 𝐴 ∈ V
3938, 38xpex 7488 . . 3 (𝐴 × 𝐴) ∈ V
4038, 39elpm2 8477 . 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 1088   = wceq 1542  wcel 2113  wral 3053  {crab 3057  wss 3841  cop 4519  cotp 4521   class class class wbr 5027   × cxp 5517  dom cdm 5519  Fun wfun 6327   Fn wfn 6328  wf 6329  cfv 6333  (class class class)co 7164  1st c1st 7705  2nd c2nd 7706  pm cpm 8431  compcco 16673  domacdoma 17385  codaccoda 17386  Arrowcarw 17387  Homachoma 17388  compaccoa 17419
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2161  ax-12 2178  ax-ext 2710  ax-rep 5151  ax-sep 5164  ax-nul 5171  ax-pow 5229  ax-pr 5293  ax-un 7473
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-reu 3060  df-rab 3062  df-v 3399  df-sbc 3680  df-csb 3789  df-dif 3844  df-un 3846  df-in 3848  df-ss 3858  df-nul 4210  df-if 4412  df-pw 4487  df-sn 4514  df-pr 4516  df-op 4520  df-ot 4522  df-uni 4794  df-iun 4880  df-br 5028  df-opab 5090  df-mpt 5108  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6291  df-fun 6335  df-fn 6336  df-f 6337  df-f1 6338  df-fo 6339  df-f1o 6340  df-fv 6341  df-ov 7167  df-oprab 7168  df-mpo 7169  df-1st 7707  df-2nd 7708  df-pm 8433  df-cat 17035  df-doma 17389  df-coda 17390  df-homa 17391  df-arw 17392  df-coa 17421
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator