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Theorem coafval 16479
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 6084 . . . . . 6 (𝑐 = 𝐶 → (Arrow‘𝑐) = (Arrow‘𝐶))
3 coafval.a . . . . . 6 𝐴 = (Arrow‘𝐶)
42, 3syl6eqr 2657 . . . . 5 (𝑐 = 𝐶 → (Arrow‘𝑐) = 𝐴)
54rabeqdv 3162 . . . . 5 (𝑐 = 𝐶 → { ∈ (Arrow‘𝑐) ∣ (coda) = (doma𝑔)} = {𝐴 ∣ (coda) = (doma𝑔)})
6 fveq2 6084 . . . . . . . . 9 (𝑐 = 𝐶 → (comp‘𝑐) = (comp‘𝐶))
7 coafval.x . . . . . . . . 9 = (comp‘𝐶)
86, 7syl6eqr 2657 . . . . . . . 8 (𝑐 = 𝐶 → (comp‘𝑐) = )
98oveqd 6540 . . . . . . 7 (𝑐 = 𝐶 → (⟨(doma𝑓), (doma𝑔)⟩(comp‘𝑐)(coda𝑔)) = (⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔)))
109oveqd 6540 . . . . . 6 (𝑐 = 𝐶 → ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩(comp‘𝑐)(coda𝑔))(2nd𝑓)) = ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓)))
1110oteq3d 4344 . . . . 5 (𝑐 = 𝐶 → ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩(comp‘𝑐)(coda𝑔))(2nd𝑓))⟩ = ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩)
124, 5, 11mpt2eq123dv 6589 . . . 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 16471 . . . 4 compa = (𝑐 ∈ Cat ↦ (𝑔 ∈ (Arrow‘𝑐), 𝑓 ∈ { ∈ (Arrow‘𝑐) ∣ (coda) = (doma𝑔)} ↦ ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩(comp‘𝑐)(coda𝑔))(2nd𝑓))⟩))
14 fvex 6094 . . . . . 6 (Arrow‘𝐶) ∈ V
153, 14eqeltri 2679 . . . . 5 𝐴 ∈ V
1615rabex 4731 . . . . 5 {𝐴 ∣ (coda) = (doma𝑔)} ∈ V
1715, 16mpt2ex 7109 . . . 4 (𝑔𝐴, 𝑓 ∈ {𝐴 ∣ (coda) = (doma𝑔)} ↦ ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩) ∈ V
1812, 13, 17fvmpt 6172 . . 3 (𝐶 ∈ Cat → (compa𝐶) = (𝑔𝐴, 𝑓 ∈ {𝐴 ∣ (coda) = (doma𝑔)} ↦ ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩))
1913dmmptss 5530 . . . . . . 7 dom compa ⊆ Cat
2019sseli 3559 . . . . . 6 (𝐶 ∈ dom compa𝐶 ∈ Cat)
2120con3i 148 . . . . 5 𝐶 ∈ Cat → ¬ 𝐶 ∈ dom compa)
22 ndmfv 6109 . . . . 5 𝐶 ∈ dom compa → (compa𝐶) = ∅)
2321, 22syl 17 . . . 4 𝐶 ∈ Cat → (compa𝐶) = ∅)
243arwrcl 16459 . . . . . . . 8 (𝑓𝐴𝐶 ∈ Cat)
2524con3i 148 . . . . . . 7 𝐶 ∈ Cat → ¬ 𝑓𝐴)
2625eq0rdv 3926 . . . . . 6 𝐶 ∈ Cat → 𝐴 = ∅)
27 eqidd 2606 . . . . . 6 𝐶 ∈ Cat → {𝐴 ∣ (coda) = (doma𝑔)} = {𝐴 ∣ (coda) = (doma𝑔)})
28 eqidd 2606 . . . . . 6 𝐶 ∈ Cat → ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩ = ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩)
2926, 27, 28mpt2eq123dv 6589 . . . . 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 6597 . . . . 5 (𝑔 ∈ ∅, 𝑓 ∈ {𝐴 ∣ (coda) = (doma𝑔)} ↦ ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩) = ∅
3129, 30syl6eq 2655 . . . 4 𝐶 ∈ Cat → (𝑔𝐴, 𝑓 ∈ {𝐴 ∣ (coda) = (doma𝑔)} ↦ ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩) = ∅)
3223, 31eqtr4d 2642 . . 3 𝐶 ∈ Cat → (compa𝐶) = (𝑔𝐴, 𝑓 ∈ {𝐴 ∣ (coda) = (doma𝑔)} ↦ ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩))
3318, 32pm2.61i 174 . 2 (compa𝐶) = (𝑔𝐴, 𝑓 ∈ {𝐴 ∣ (coda) = (doma𝑔)} ↦ ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩)
341, 33eqtri 2627 1 · = (𝑔𝐴, 𝑓 ∈ {𝐴 ∣ (coda) = (doma𝑔)} ↦ ⟨(doma𝑓), (coda𝑔), ((2nd𝑔)(⟨(doma𝑓), (doma𝑔)⟩ (coda𝑔))(2nd𝑓))⟩)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3   = wceq 1474  wcel 1975  {crab 2895  Vcvv 3168  c0 3869  cop 4126  cotp 4128  dom cdm 5024  cfv 5786  (class class class)co 6523  cmpt2 6525  2nd c2nd 7031  compcco 15722  Catccat 16090  domacdoma 16435  codaccoda 16436  Arrowcarw 16437  compaccoa 16469
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585  ax-rep 4689  ax-sep 4699  ax-nul 4708  ax-pow 4760  ax-pr 4824  ax-un 6820
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2457  df-mo 2458  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-ne 2777  df-ral 2896  df-rex 2897  df-reu 2898  df-rab 2900  df-v 3170  df-sbc 3398  df-csb 3495  df-dif 3538  df-un 3540  df-in 3542  df-ss 3549  df-nul 3870  df-if 4032  df-pw 4105  df-sn 4121  df-pr 4123  df-op 4127  df-ot 4129  df-uni 4363  df-iun 4447  df-br 4574  df-opab 4634  df-mpt 4635  df-id 4939  df-xp 5030  df-rel 5031  df-cnv 5032  df-co 5033  df-dm 5034  df-rn 5035  df-res 5036  df-ima 5037  df-iota 5750  df-fun 5788  df-fn 5789  df-f 5790  df-f1 5791  df-fo 5792  df-f1o 5793  df-fv 5794  df-ov 6526  df-oprab 6527  df-mpt2 6528  df-1st 7032  df-2nd 7033  df-arw 16442  df-coa 16471
This theorem is referenced by:  eldmcoa  16480  dmcoass  16481  coaval  16483  coapm  16486
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