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Theorem coafval 16886
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 6340 . . . . . 6 (𝑐 = 𝐶 → (Arrow‘𝑐) = (Arrow‘𝐶))
3 coafval.a . . . . . 6 𝐴 = (Arrow‘𝐶)
42, 3syl6eqr 2800 . . . . 5 (𝑐 = 𝐶 → (Arrow‘𝑐) = 𝐴)
54rabeqdv 3322 . . . . 5 (𝑐 = 𝐶 → { ∈ (Arrow‘𝑐) ∣ (coda) = (doma𝑔)} = {𝐴 ∣ (coda) = (doma𝑔)})
6 fveq2 6340 . . . . . . . . 9 (𝑐 = 𝐶 → (comp‘𝑐) = (comp‘𝐶))
7 coafval.x . . . . . . . . 9 = (comp‘𝐶)
86, 7syl6eqr 2800 . . . . . . . 8 (𝑐 = 𝐶 → (comp‘𝑐) = )
98oveqd 6818 . . . . . . 7 (𝑐 = 𝐶 → (⟨(doma𝑓), (doma𝑔)⟩(comp‘𝑐)(coda𝑔)) = (⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔)))
109oveqd 6818 . . . . . 6 (𝑐 = 𝐶 → ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩(comp‘𝑐)(coda𝑔))(2nd𝑓)) = ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓)))
1110oteq3d 4555 . . . . 5 (𝑐 = 𝐶 → ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩(comp‘𝑐)(coda𝑔))(2nd𝑓))⟩ = ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩)
124, 5, 11mpt2eq123dv 6870 . . . 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 16878 . . . 4 compa = (𝑐 ∈ Cat ↦ (𝑔 ∈ (Arrow‘𝑐), 𝑓 ∈ { ∈ (Arrow‘𝑐) ∣ (coda) = (doma𝑔)} ↦ ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩(comp‘𝑐)(coda𝑔))(2nd𝑓))⟩))
14 fvex 6350 . . . . . 6 (Arrow‘𝐶) ∈ V
153, 14eqeltri 2823 . . . . 5 𝐴 ∈ V
1615rabex 4952 . . . . 5 {𝐴 ∣ (coda) = (doma𝑔)} ∈ V
1715, 16mpt2ex 7403 . . . 4 (𝑔𝐴, 𝑓 ∈ {𝐴 ∣ (coda) = (doma𝑔)} ↦ ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩) ∈ V
1812, 13, 17fvmpt 6432 . . 3 (𝐶 ∈ Cat → (compa𝐶) = (𝑔𝐴, 𝑓 ∈ {𝐴 ∣ (coda) = (doma𝑔)} ↦ ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩))
1913dmmptss 5780 . . . . . . 7 dom compa ⊆ Cat
2019sseli 3728 . . . . . 6 (𝐶 ∈ dom compa𝐶 ∈ Cat)
2120con3i 150 . . . . 5 𝐶 ∈ Cat → ¬ 𝐶 ∈ dom compa)
22 ndmfv 6367 . . . . 5 𝐶 ∈ dom compa → (compa𝐶) = ∅)
2321, 22syl 17 . . . 4 𝐶 ∈ Cat → (compa𝐶) = ∅)
243arwrcl 16866 . . . . . . . 8 (𝑓𝐴𝐶 ∈ Cat)
2524con3i 150 . . . . . . 7 𝐶 ∈ Cat → ¬ 𝑓𝐴)
2625eq0rdv 4110 . . . . . 6 𝐶 ∈ Cat → 𝐴 = ∅)
27 eqidd 2749 . . . . . 6 𝐶 ∈ Cat → {𝐴 ∣ (coda) = (doma𝑔)} = {𝐴 ∣ (coda) = (doma𝑔)})
28 eqidd 2749 . . . . . 6 𝐶 ∈ Cat → ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩ = ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩)
2926, 27, 28mpt2eq123dv 6870 . . . . 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 6878 . . . . 5 (𝑔 ∈ ∅, 𝑓 ∈ {𝐴 ∣ (coda) = (doma𝑔)} ↦ ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩) = ∅
3129, 30syl6eq 2798 . . . 4 𝐶 ∈ Cat → (𝑔𝐴, 𝑓 ∈ {𝐴 ∣ (coda) = (doma𝑔)} ↦ ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩) = ∅)
3223, 31eqtr4d 2785 . . 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 2770 1 · = (𝑔𝐴, 𝑓 ∈ {𝐴 ∣ (coda) = (doma𝑔)} ↦ ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1620  wcel 2127  {crab 3042  Vcvv 3328  c0 4046  cop 4315  cotp 4317  dom cdm 5254  cfv 6037  (class class class)co 6801  cmpt2 6803  2nd c2nd 7320  compcco 16126  Catccat 16497  domacdoma 16842  codaccoda 16843  Arrowcarw 16844  compaccoa 16876
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-rep 4911  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-rex 3044  df-reu 3045  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-op 4316  df-ot 4318  df-uni 4577  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-id 5162  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fo 6043  df-f1o 6044  df-fv 6045  df-ov 6804  df-oprab 6805  df-mpt2 6806  df-1st 7321  df-2nd 7322  df-arw 16849  df-coa 16878
This theorem is referenced by:  eldmcoa  16887  dmcoass  16888  coaval  16890  coapm  16893
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