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

Theorem dmcoass 18111
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 2737 . . . 4 (comp‘𝐶) = (comp‘𝐶)
41, 2, 3coafval 18109 . . 3 · = (𝑔𝐴, 𝑓 ∈ {𝐴 ∣ (coda) = (doma𝑔)} ↦ ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩(comp‘𝐶)(coda𝑔))(2nd𝑓))⟩)
54dmmpossx 8091 . 2 dom · 𝑔𝐴 ({𝑔} × {𝐴 ∣ (coda) = (doma𝑔)})
6 iunss 5045 . . 3 ( 𝑔𝐴 ({𝑔} × {𝐴 ∣ (coda) = (doma𝑔)}) ⊆ (𝐴 × 𝐴) ↔ ∀𝑔𝐴 ({𝑔} × {𝐴 ∣ (coda) = (doma𝑔)}) ⊆ (𝐴 × 𝐴))
7 snssi 4808 . . . 4 (𝑔𝐴 → {𝑔} ⊆ 𝐴)
8 ssrab2 4080 . . . 4 {𝐴 ∣ (coda) = (doma𝑔)} ⊆ 𝐴
9 xpss12 5700 . . . 4 (({𝑔} ⊆ 𝐴 ∧ {𝐴 ∣ (coda) = (doma𝑔)} ⊆ 𝐴) → ({𝑔} × {𝐴 ∣ (coda) = (doma𝑔)}) ⊆ (𝐴 × 𝐴))
107, 8, 9sylancl 586 . . 3 (𝑔𝐴 → ({𝑔} × {𝐴 ∣ (coda) = (doma𝑔)}) ⊆ (𝐴 × 𝐴))
116, 10mprgbir 3068 . 2 𝑔𝐴 ({𝑔} × {𝐴 ∣ (coda) = (doma𝑔)}) ⊆ (𝐴 × 𝐴)
125, 11sstri 3993 1 dom · ⊆ (𝐴 × 𝐴)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  wcel 2108  {crab 3436  wss 3951  {csn 4626  cop 4632  cotp 4634   ciun 4991   × cxp 5683  dom cdm 5685  cfv 6561  (class class class)co 7431  2nd c2nd 8013  compcco 17309  domacdoma 18065  codaccoda 18066  Arrowcarw 18067  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-arw 18072  df-coa 18101
This theorem is referenced by:  coapm  18116
  Copyright terms: Public domain W3C validator