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Theorem coafval 18001
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 6881 . . . . . 6 (𝑐 = 𝐶 → (Arrow‘𝑐) = (Arrow‘𝐶))
3 coafval.a . . . . . 6 𝐴 = (Arrow‘𝐶)
42, 3eqtr4di 2791 . . . . 5 (𝑐 = 𝐶 → (Arrow‘𝑐) = 𝐴)
54rabeqdv 3448 . . . . 5 (𝑐 = 𝐶 → { ∈ (Arrow‘𝑐) ∣ (coda) = (doma𝑔)} = {𝐴 ∣ (coda) = (doma𝑔)})
6 fveq2 6881 . . . . . . . . 9 (𝑐 = 𝐶 → (comp‘𝑐) = (comp‘𝐶))
7 coafval.x . . . . . . . . 9 = (comp‘𝐶)
86, 7eqtr4di 2791 . . . . . . . 8 (𝑐 = 𝐶 → (comp‘𝑐) = )
98oveqd 7413 . . . . . . 7 (𝑐 = 𝐶 → (⟨(doma𝑓), (doma𝑔)⟩(comp‘𝑐)(coda𝑔)) = (⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔)))
109oveqd 7413 . . . . . 6 (𝑐 = 𝐶 → ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩(comp‘𝑐)(coda𝑔))(2nd𝑓)) = ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓)))
1110oteq3d 4883 . . . . 5 (𝑐 = 𝐶 → ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩(comp‘𝑐)(coda𝑔))(2nd𝑓))⟩ = ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩)
124, 5, 11mpoeq123dv 7471 . . . 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 17993 . . . 4 compa = (𝑐 ∈ Cat ↦ (𝑔 ∈ (Arrow‘𝑐), 𝑓 ∈ { ∈ (Arrow‘𝑐) ∣ (coda) = (doma𝑔)} ↦ ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩(comp‘𝑐)(coda𝑔))(2nd𝑓))⟩))
143fvexi 6895 . . . . 5 𝐴 ∈ V
1514rabex 5328 . . . . 5 {𝐴 ∣ (coda) = (doma𝑔)} ∈ V
1614, 15mpoex 8053 . . . 4 (𝑔𝐴, 𝑓 ∈ {𝐴 ∣ (coda) = (doma𝑔)} ↦ ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩) ∈ V
1712, 13, 16fvmpt 6987 . . 3 (𝐶 ∈ Cat → (compa𝐶) = (𝑔𝐴, 𝑓 ∈ {𝐴 ∣ (coda) = (doma𝑔)} ↦ ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩))
1813fvmptndm 7017 . . . 4 𝐶 ∈ Cat → (compa𝐶) = ∅)
193arwrcl 17981 . . . . . . . 8 (𝑓𝐴𝐶 ∈ Cat)
2019con3i 154 . . . . . . 7 𝐶 ∈ Cat → ¬ 𝑓𝐴)
2120eq0rdv 4402 . . . . . 6 𝐶 ∈ Cat → 𝐴 = ∅)
22 eqidd 2734 . . . . . 6 𝐶 ∈ Cat → {𝐴 ∣ (coda) = (doma𝑔)} = {𝐴 ∣ (coda) = (doma𝑔)})
23 eqidd 2734 . . . . . 6 𝐶 ∈ Cat → ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩ = ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩)
2421, 22, 23mpoeq123dv 7471 . . . . 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))⟩))
25 mpo0 7481 . . . . 5 (𝑔 ∈ ∅, 𝑓 ∈ {𝐴 ∣ (coda) = (doma𝑔)} ↦ ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩) = ∅
2624, 25eqtrdi 2789 . . . 4 𝐶 ∈ Cat → (𝑔𝐴, 𝑓 ∈ {𝐴 ∣ (coda) = (doma𝑔)} ↦ ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩) = ∅)
2718, 26eqtr4d 2776 . . 3 𝐶 ∈ Cat → (compa𝐶) = (𝑔𝐴, 𝑓 ∈ {𝐴 ∣ (coda) = (doma𝑔)} ↦ ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩))
2817, 27pm2.61i 182 . 2 (compa𝐶) = (𝑔𝐴, 𝑓 ∈ {𝐴 ∣ (coda) = (doma𝑔)} ↦ ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩)
291, 28eqtri 2761 1 · = (𝑔𝐴, 𝑓 ∈ {𝐴 ∣ (coda) = (doma𝑔)} ↦ ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1542  wcel 2107  {crab 3433  c0 4320  cop 4630  cotp 4632  cfv 6535  (class class class)co 7396  cmpo 7398  2nd c2nd 7961  compcco 17196  Catccat 17595  domacdoma 17957  codaccoda 17958  Arrowcarw 17959  compaccoa 17991
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5281  ax-sep 5295  ax-nul 5302  ax-pow 5359  ax-pr 5423  ax-un 7712
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-ot 4633  df-uni 4905  df-iun 4995  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6487  df-fun 6537  df-fn 6538  df-f 6539  df-f1 6540  df-fo 6541  df-f1o 6542  df-fv 6543  df-ov 7399  df-oprab 7400  df-mpo 7401  df-1st 7962  df-2nd 7963  df-arw 17964  df-coa 17993
This theorem is referenced by:  eldmcoa  18002  dmcoass  18003  coaval  18005  coapm  18008
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