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

Theorem dmcoass 17069
 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 2826 . . . 4 (comp‘𝐶) = (comp‘𝐶)
41, 2, 3coafval 17067 . . 3 · = (𝑔𝐴, 𝑓 ∈ {𝐴 ∣ (coda) = (doma𝑔)} ↦ ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩(comp‘𝐶)(coda𝑔))(2nd𝑓))⟩)
54dmmpt2ssx 7499 . 2 dom · 𝑔𝐴 ({𝑔} × {𝐴 ∣ (coda) = (doma𝑔)})
6 iunss 4782 . . 3 ( 𝑔𝐴 ({𝑔} × {𝐴 ∣ (coda) = (doma𝑔)}) ⊆ (𝐴 × 𝐴) ↔ ∀𝑔𝐴 ({𝑔} × {𝐴 ∣ (coda) = (doma𝑔)}) ⊆ (𝐴 × 𝐴))
7 snssi 4558 . . . 4 (𝑔𝐴 → {𝑔} ⊆ 𝐴)
8 ssrab2 3913 . . . 4 {𝐴 ∣ (coda) = (doma𝑔)} ⊆ 𝐴
9 xpss12 5358 . . . 4 (({𝑔} ⊆ 𝐴 ∧ {𝐴 ∣ (coda) = (doma𝑔)} ⊆ 𝐴) → ({𝑔} × {𝐴 ∣ (coda) = (doma𝑔)}) ⊆ (𝐴 × 𝐴))
107, 8, 9sylancl 582 . . 3 (𝑔𝐴 → ({𝑔} × {𝐴 ∣ (coda) = (doma𝑔)}) ⊆ (𝐴 × 𝐴))
116, 10mprgbir 3137 . 2 𝑔𝐴 ({𝑔} × {𝐴 ∣ (coda) = (doma𝑔)}) ⊆ (𝐴 × 𝐴)
125, 11sstri 3837 1 dom · ⊆ (𝐴 × 𝐴)
 Colors of variables: wff setvar class Syntax hints:   = wceq 1658   ∈ wcel 2166  {crab 3122   ⊆ wss 3799  {csn 4398  ⟨cop 4404  ⟨cotp 4406  ∪ ciun 4741   × cxp 5341  dom cdm 5343  ‘cfv 6124  (class class class)co 6906  2nd c2nd 7428  compcco 16318  domacdoma 17023  codaccoda 17024  Arrowcarw 17025  compaccoa 17057 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2804  ax-rep 4995  ax-sep 5006  ax-nul 5014  ax-pow 5066  ax-pr 5128  ax-un 7210 This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2606  df-eu 2641  df-clab 2813  df-cleq 2819  df-clel 2822  df-nfc 2959  df-ne 3001  df-ral 3123  df-rex 3124  df-reu 3125  df-rab 3127  df-v 3417  df-sbc 3664  df-csb 3759  df-dif 3802  df-un 3804  df-in 3806  df-ss 3813  df-nul 4146  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-op 4405  df-ot 4407  df-uni 4660  df-iun 4743  df-br 4875  df-opab 4937  df-mpt 4954  df-id 5251  df-xp 5349  df-rel 5350  df-cnv 5351  df-co 5352  df-dm 5353  df-rn 5354  df-res 5355  df-ima 5356  df-iota 6087  df-fun 6126  df-fn 6127  df-f 6128  df-f1 6129  df-fo 6130  df-f1o 6131  df-fv 6132  df-ov 6909  df-oprab 6910  df-mpt2 6911  df-1st 7429  df-2nd 7430  df-arw 17030  df-coa 17059 This theorem is referenced by:  coapm  17074
 Copyright terms: Public domain W3C validator