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Theorem coafval 16654
Description: The value of the composition of arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
coafval.o · = (compa𝐶)
coafval.a 𝐴 = (Arrow‘𝐶)
coafval.x = (comp‘𝐶)
Assertion
Ref Expression
coafval · = (𝑔𝐴, 𝑓 ∈ {𝐴 ∣ (coda) = (doma𝑔)} ↦ ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩)
Distinct variable groups:   𝑓,𝑔,,𝐴   𝐶,𝑓,𝑔,
Allowed substitution hints:   (𝑓,𝑔,)   · (𝑓,𝑔,)

Proof of Theorem coafval
Dummy variable 𝑐 is distinct from all other variables.
StepHypRef Expression
1 coafval.o . 2 · = (compa𝐶)
2 fveq2 6158 . . . . . 6 (𝑐 = 𝐶 → (Arrow‘𝑐) = (Arrow‘𝐶))
3 coafval.a . . . . . 6 𝐴 = (Arrow‘𝐶)
42, 3syl6eqr 2673 . . . . 5 (𝑐 = 𝐶 → (Arrow‘𝑐) = 𝐴)
54rabeqdv 3184 . . . . 5 (𝑐 = 𝐶 → { ∈ (Arrow‘𝑐) ∣ (coda) = (doma𝑔)} = {𝐴 ∣ (coda) = (doma𝑔)})
6 fveq2 6158 . . . . . . . . 9 (𝑐 = 𝐶 → (comp‘𝑐) = (comp‘𝐶))
7 coafval.x . . . . . . . . 9 = (comp‘𝐶)
86, 7syl6eqr 2673 . . . . . . . 8 (𝑐 = 𝐶 → (comp‘𝑐) = )
98oveqd 6632 . . . . . . 7 (𝑐 = 𝐶 → (⟨(doma𝑓), (doma𝑔)⟩(comp‘𝑐)(coda𝑔)) = (⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔)))
109oveqd 6632 . . . . . 6 (𝑐 = 𝐶 → ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩(comp‘𝑐)(coda𝑔))(2nd𝑓)) = ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓)))
1110oteq3d 4391 . . . . 5 (𝑐 = 𝐶 → ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩(comp‘𝑐)(coda𝑔))(2nd𝑓))⟩ = ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩)
124, 5, 11mpt2eq123dv 6682 . . . 4 (𝑐 = 𝐶 → (𝑔 ∈ (Arrow‘𝑐), 𝑓 ∈ { ∈ (Arrow‘𝑐) ∣ (coda) = (doma𝑔)} ↦ ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩(comp‘𝑐)(coda𝑔))(2nd𝑓))⟩) = (𝑔𝐴, 𝑓 ∈ {𝐴 ∣ (coda) = (doma𝑔)} ↦ ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩))
13 df-coa 16646 . . . 4 compa = (𝑐 ∈ Cat ↦ (𝑔 ∈ (Arrow‘𝑐), 𝑓 ∈ { ∈ (Arrow‘𝑐) ∣ (coda) = (doma𝑔)} ↦ ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩(comp‘𝑐)(coda𝑔))(2nd𝑓))⟩))
14 fvex 6168 . . . . . 6 (Arrow‘𝐶) ∈ V
153, 14eqeltri 2694 . . . . 5 𝐴 ∈ V
1615rabex 4783 . . . . 5 {𝐴 ∣ (coda) = (doma𝑔)} ∈ V
1715, 16mpt2ex 7207 . . . 4 (𝑔𝐴, 𝑓 ∈ {𝐴 ∣ (coda) = (doma𝑔)} ↦ ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩) ∈ V
1812, 13, 17fvmpt 6249 . . 3 (𝐶 ∈ Cat → (compa𝐶) = (𝑔𝐴, 𝑓 ∈ {𝐴 ∣ (coda) = (doma𝑔)} ↦ ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩))
1913dmmptss 5600 . . . . . . 7 dom compa ⊆ Cat
2019sseli 3584 . . . . . 6 (𝐶 ∈ dom compa𝐶 ∈ Cat)
2120con3i 150 . . . . 5 𝐶 ∈ Cat → ¬ 𝐶 ∈ dom compa)
22 ndmfv 6185 . . . . 5 𝐶 ∈ dom compa → (compa𝐶) = ∅)
2321, 22syl 17 . . . 4 𝐶 ∈ Cat → (compa𝐶) = ∅)
243arwrcl 16634 . . . . . . . 8 (𝑓𝐴𝐶 ∈ Cat)
2524con3i 150 . . . . . . 7 𝐶 ∈ Cat → ¬ 𝑓𝐴)
2625eq0rdv 3957 . . . . . 6 𝐶 ∈ Cat → 𝐴 = ∅)
27 eqidd 2622 . . . . . 6 𝐶 ∈ Cat → {𝐴 ∣ (coda) = (doma𝑔)} = {𝐴 ∣ (coda) = (doma𝑔)})
28 eqidd 2622 . . . . . 6 𝐶 ∈ Cat → ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩ = ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩)
2926, 27, 28mpt2eq123dv 6682 . . . . 5 𝐶 ∈ Cat → (𝑔𝐴, 𝑓 ∈ {𝐴 ∣ (coda) = (doma𝑔)} ↦ ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩) = (𝑔 ∈ ∅, 𝑓 ∈ {𝐴 ∣ (coda) = (doma𝑔)} ↦ ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩))
30 mpt20 6690 . . . . 5 (𝑔 ∈ ∅, 𝑓 ∈ {𝐴 ∣ (coda) = (doma𝑔)} ↦ ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩) = ∅
3129, 30syl6eq 2671 . . . 4 𝐶 ∈ Cat → (𝑔𝐴, 𝑓 ∈ {𝐴 ∣ (coda) = (doma𝑔)} ↦ ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩) = ∅)
3223, 31eqtr4d 2658 . . 3 𝐶 ∈ Cat → (compa𝐶) = (𝑔𝐴, 𝑓 ∈ {𝐴 ∣ (coda) = (doma𝑔)} ↦ ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩))
3318, 32pm2.61i 176 . 2 (compa𝐶) = (𝑔𝐴, 𝑓 ∈ {𝐴 ∣ (coda) = (doma𝑔)} ↦ ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩)
341, 33eqtri 2643 1 · = (𝑔𝐴, 𝑓 ∈ {𝐴 ∣ (coda) = (doma𝑔)} ↦ ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1480  wcel 1987  {crab 2912  Vcvv 3190  c0 3897  cop 4161  cotp 4163  dom cdm 5084  cfv 5857  (class class class)co 6615  cmpt2 6617  2nd c2nd 7127  compcco 15893  Catccat 16265  domacdoma 16610  codaccoda 16611  Arrowcarw 16612  compaccoa 16644
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4741  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-reu 2915  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-ot 4164  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-1st 7128  df-2nd 7129  df-arw 16617  df-coa 16646
This theorem is referenced by:  eldmcoa  16655  dmcoass  16656  coaval  16658  coapm  16661
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