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Theorem dmcoass 18026
Description: The domain of composition is a collection of pairs of arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
coafval.o · = (compa𝐶)
coafval.a 𝐴 = (Arrow‘𝐶)
Assertion
Ref Expression
dmcoass dom · ⊆ (𝐴 × 𝐴)

Proof of Theorem dmcoass
Dummy variables 𝑓 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 coafval.o . . . 4 · = (compa𝐶)
2 coafval.a . . . 4 𝐴 = (Arrow‘𝐶)
3 eqid 2731 . . . 4 (comp‘𝐶) = (comp‘𝐶)
41, 2, 3coafval 18024 . . 3 · = (𝑔𝐴, 𝑓 ∈ {𝐴 ∣ (coda) = (doma𝑔)} ↦ ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩(comp‘𝐶)(coda𝑔))(2nd𝑓))⟩)
54dmmpossx 8056 . 2 dom · 𝑔𝐴 ({𝑔} × {𝐴 ∣ (coda) = (doma𝑔)})
6 iunss 5048 . . 3 ( 𝑔𝐴 ({𝑔} × {𝐴 ∣ (coda) = (doma𝑔)}) ⊆ (𝐴 × 𝐴) ↔ ∀𝑔𝐴 ({𝑔} × {𝐴 ∣ (coda) = (doma𝑔)}) ⊆ (𝐴 × 𝐴))
7 snssi 4811 . . . 4 (𝑔𝐴 → {𝑔} ⊆ 𝐴)
8 ssrab2 4077 . . . 4 {𝐴 ∣ (coda) = (doma𝑔)} ⊆ 𝐴
9 xpss12 5691 . . . 4 (({𝑔} ⊆ 𝐴 ∧ {𝐴 ∣ (coda) = (doma𝑔)} ⊆ 𝐴) → ({𝑔} × {𝐴 ∣ (coda) = (doma𝑔)}) ⊆ (𝐴 × 𝐴))
107, 8, 9sylancl 585 . . 3 (𝑔𝐴 → ({𝑔} × {𝐴 ∣ (coda) = (doma𝑔)}) ⊆ (𝐴 × 𝐴))
116, 10mprgbir 3067 . 2 𝑔𝐴 ({𝑔} × {𝐴 ∣ (coda) = (doma𝑔)}) ⊆ (𝐴 × 𝐴)
125, 11sstri 3991 1 dom · ⊆ (𝐴 × 𝐴)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  wcel 2105  {crab 3431  wss 3948  {csn 4628  cop 4634  cotp 4636   ciun 4997   × cxp 5674  dom cdm 5676  cfv 6543  (class class class)co 7412  2nd c2nd 7978  compcco 17216  domacdoma 17980  codaccoda 17981  Arrowcarw 17982  compaccoa 18014
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7729
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-ot 4637  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-1st 7979  df-2nd 7980  df-arw 17987  df-coa 18016
This theorem is referenced by:  coapm  18031
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