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

Theorem dmcoass 18020
Description: The domain of composition is a collection of pairs of arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
coafval.o Β· = (compaβ€˜πΆ)
coafval.a 𝐴 = (Arrowβ€˜πΆ)
Assertion
Ref Expression
dmcoass dom Β· βŠ† (𝐴 Γ— 𝐴)

Proof of Theorem dmcoass
Dummy variables 𝑓 𝑔 β„Ž are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 coafval.o . . . 4 Β· = (compaβ€˜πΆ)
2 coafval.a . . . 4 𝐴 = (Arrowβ€˜πΆ)
3 eqid 2732 . . . 4 (compβ€˜πΆ) = (compβ€˜πΆ)
41, 2, 3coafval 18018 . . 3 Β· = (𝑔 ∈ 𝐴, 𝑓 ∈ {β„Ž ∈ 𝐴 ∣ (codaβ€˜β„Ž) = (domaβ€˜π‘”)} ↦ ⟨(domaβ€˜π‘“), (codaβ€˜π‘”), ((2nd β€˜π‘”)(⟨(domaβ€˜π‘“), (domaβ€˜π‘”)⟩(compβ€˜πΆ)(codaβ€˜π‘”))(2nd β€˜π‘“))⟩)
54dmmpossx 8054 . 2 dom Β· βŠ† βˆͺ 𝑔 ∈ 𝐴 ({𝑔} Γ— {β„Ž ∈ 𝐴 ∣ (codaβ€˜β„Ž) = (domaβ€˜π‘”)})
6 iunss 5048 . . 3 (βˆͺ 𝑔 ∈ 𝐴 ({𝑔} Γ— {β„Ž ∈ 𝐴 ∣ (codaβ€˜β„Ž) = (domaβ€˜π‘”)}) βŠ† (𝐴 Γ— 𝐴) ↔ βˆ€π‘” ∈ 𝐴 ({𝑔} Γ— {β„Ž ∈ 𝐴 ∣ (codaβ€˜β„Ž) = (domaβ€˜π‘”)}) βŠ† (𝐴 Γ— 𝐴))
7 snssi 4811 . . . 4 (𝑔 ∈ 𝐴 β†’ {𝑔} βŠ† 𝐴)
8 ssrab2 4077 . . . 4 {β„Ž ∈ 𝐴 ∣ (codaβ€˜β„Ž) = (domaβ€˜π‘”)} βŠ† 𝐴
9 xpss12 5691 . . . 4 (({𝑔} βŠ† 𝐴 ∧ {β„Ž ∈ 𝐴 ∣ (codaβ€˜β„Ž) = (domaβ€˜π‘”)} βŠ† 𝐴) β†’ ({𝑔} Γ— {β„Ž ∈ 𝐴 ∣ (codaβ€˜β„Ž) = (domaβ€˜π‘”)}) βŠ† (𝐴 Γ— 𝐴))
107, 8, 9sylancl 586 . . 3 (𝑔 ∈ 𝐴 β†’ ({𝑔} Γ— {β„Ž ∈ 𝐴 ∣ (codaβ€˜β„Ž) = (domaβ€˜π‘”)}) βŠ† (𝐴 Γ— 𝐴))
116, 10mprgbir 3068 . 2 βˆͺ 𝑔 ∈ 𝐴 ({𝑔} Γ— {β„Ž ∈ 𝐴 ∣ (codaβ€˜β„Ž) = (domaβ€˜π‘”)}) βŠ† (𝐴 Γ— 𝐴)
125, 11sstri 3991 1 dom Β· βŠ† (𝐴 Γ— 𝐴)
Colors of variables: wff setvar class
Syntax hints:   = wceq 1541   ∈ wcel 2106  {crab 3432   βŠ† wss 3948  {csn 4628  βŸ¨cop 4634  βŸ¨cotp 4636  βˆͺ ciun 4997   Γ— cxp 5674  dom cdm 5676  β€˜cfv 6543  (class class class)co 7411  2nd c2nd 7976  compcco 17213  domacdoma 17974  codaccoda 17975  Arrowcarw 17976  compaccoa 18008
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  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 7414  df-oprab 7415  df-mpo 7416  df-1st 7977  df-2nd 7978  df-arw 17981  df-coa 18010
This theorem is referenced by:  coapm  18025
  Copyright terms: Public domain W3C validator