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Theorem dmcoass 18119
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 2734 . . . 4 (comp‘𝐶) = (comp‘𝐶)
41, 2, 3coafval 18117 . . 3 · = (𝑔𝐴, 𝑓 ∈ {𝐴 ∣ (coda) = (doma𝑔)} ↦ ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩(comp‘𝐶)(coda𝑔))(2nd𝑓))⟩)
54dmmpossx 8089 . 2 dom · 𝑔𝐴 ({𝑔} × {𝐴 ∣ (coda) = (doma𝑔)})
6 iunss 5049 . . 3 ( 𝑔𝐴 ({𝑔} × {𝐴 ∣ (coda) = (doma𝑔)}) ⊆ (𝐴 × 𝐴) ↔ ∀𝑔𝐴 ({𝑔} × {𝐴 ∣ (coda) = (doma𝑔)}) ⊆ (𝐴 × 𝐴))
7 snssi 4812 . . . 4 (𝑔𝐴 → {𝑔} ⊆ 𝐴)
8 ssrab2 4089 . . . 4 {𝐴 ∣ (coda) = (doma𝑔)} ⊆ 𝐴
9 xpss12 5703 . . . 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 3065 . 2 𝑔𝐴 ({𝑔} × {𝐴 ∣ (coda) = (doma𝑔)}) ⊆ (𝐴 × 𝐴)
125, 11sstri 4004 1 dom · ⊆ (𝐴 × 𝐴)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1536  wcel 2105  {crab 3432  wss 3962  {csn 4630  cop 4636  cotp 4638   ciun 4995   × cxp 5686  dom cdm 5688  cfv 6562  (class class class)co 7430  2nd c2nd 8011  compcco 17309  domacdoma 18073  codaccoda 18074  Arrowcarw 18075  compaccoa 18107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-ot 4639  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-1st 8012  df-2nd 8013  df-arw 18080  df-coa 18109
This theorem is referenced by:  coapm  18124
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