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Theorem dmcoass 17304
 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 2821 . . . 4 (comp‘𝐶) = (comp‘𝐶)
41, 2, 3coafval 17302 . . 3 · = (𝑔𝐴, 𝑓 ∈ {𝐴 ∣ (coda) = (doma𝑔)} ↦ ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩(comp‘𝐶)(coda𝑔))(2nd𝑓))⟩)
54dmmpossx 7739 . 2 dom · 𝑔𝐴 ({𝑔} × {𝐴 ∣ (coda) = (doma𝑔)})
6 iunss 4942 . . 3 ( 𝑔𝐴 ({𝑔} × {𝐴 ∣ (coda) = (doma𝑔)}) ⊆ (𝐴 × 𝐴) ↔ ∀𝑔𝐴 ({𝑔} × {𝐴 ∣ (coda) = (doma𝑔)}) ⊆ (𝐴 × 𝐴))
7 snssi 4714 . . . 4 (𝑔𝐴 → {𝑔} ⊆ 𝐴)
8 ssrab2 4032 . . . 4 {𝐴 ∣ (coda) = (doma𝑔)} ⊆ 𝐴
9 xpss12 5543 . . . 4 (({𝑔} ⊆ 𝐴 ∧ {𝐴 ∣ (coda) = (doma𝑔)} ⊆ 𝐴) → ({𝑔} × {𝐴 ∣ (coda) = (doma𝑔)}) ⊆ (𝐴 × 𝐴))
107, 8, 9sylancl 589 . . 3 (𝑔𝐴 → ({𝑔} × {𝐴 ∣ (coda) = (doma𝑔)}) ⊆ (𝐴 × 𝐴))
116, 10mprgbir 3141 . 2 𝑔𝐴 ({𝑔} × {𝐴 ∣ (coda) = (doma𝑔)}) ⊆ (𝐴 × 𝐴)
125, 11sstri 3952 1 dom · ⊆ (𝐴 × 𝐴)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ∈ wcel 2115  {crab 3130   ⊆ wss 3910  {csn 4540  ⟨cop 4546  ⟨cotp 4548  ∪ ciun 4892   × cxp 5526  dom cdm 5528  ‘cfv 6328  (class class class)co 7130  2nd c2nd 7663  compcco 16555  domacdoma 17258  codaccoda 17259  Arrowcarw 17260  compaccoa 17292 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-op 4547  df-ot 4549  df-uni 4812  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-id 5433  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-ov 7133  df-oprab 7134  df-mpo 7135  df-1st 7664  df-2nd 7665  df-arw 17265  df-coa 17294 This theorem is referenced by:  coapm  17309
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