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Theorem coa2 16542
 Description: The morphism part of arrow composition. (Contributed by Mario Carneiro, 11-Jan-2017.)
Hypotheses
Ref Expression
homdmcoa.o · = (compa𝐶)
homdmcoa.h 𝐻 = (Homa𝐶)
homdmcoa.f (𝜑𝐹 ∈ (𝑋𝐻𝑌))
homdmcoa.g (𝜑𝐺 ∈ (𝑌𝐻𝑍))
coaval.x = (comp‘𝐶)
Assertion
Ref Expression
coa2 (𝜑 → (2nd ‘(𝐺 · 𝐹)) = ((2nd𝐺)(⟨𝑋, 𝑌 𝑍)(2nd𝐹)))

Proof of Theorem coa2
StepHypRef Expression
1 homdmcoa.o . . . 4 · = (compa𝐶)
2 homdmcoa.h . . . 4 𝐻 = (Homa𝐶)
3 homdmcoa.f . . . 4 (𝜑𝐹 ∈ (𝑋𝐻𝑌))
4 homdmcoa.g . . . 4 (𝜑𝐺 ∈ (𝑌𝐻𝑍))
5 coaval.x . . . 4 = (comp‘𝐶)
61, 2, 3, 4, 5coaval 16541 . . 3 (𝜑 → (𝐺 · 𝐹) = ⟨𝑋, 𝑍, ((2nd𝐺)(⟨𝑋, 𝑌 𝑍)(2nd𝐹))⟩)
76fveq2d 6107 . 2 (𝜑 → (2nd ‘(𝐺 · 𝐹)) = (2nd ‘⟨𝑋, 𝑍, ((2nd𝐺)(⟨𝑋, 𝑌 𝑍)(2nd𝐹))⟩))
8 ovex 6577 . . 3 ((2nd𝐺)(⟨𝑋, 𝑌 𝑍)(2nd𝐹)) ∈ V
9 ot3rdg 7075 . . 3 (((2nd𝐺)(⟨𝑋, 𝑌 𝑍)(2nd𝐹)) ∈ V → (2nd ‘⟨𝑋, 𝑍, ((2nd𝐺)(⟨𝑋, 𝑌 𝑍)(2nd𝐹))⟩) = ((2nd𝐺)(⟨𝑋, 𝑌 𝑍)(2nd𝐹)))
108, 9ax-mp 5 . 2 (2nd ‘⟨𝑋, 𝑍, ((2nd𝐺)(⟨𝑋, 𝑌 𝑍)(2nd𝐹))⟩) = ((2nd𝐺)(⟨𝑋, 𝑌 𝑍)(2nd𝐹))
117, 10syl6eq 2660 1 (𝜑 → (2nd ‘(𝐺 · 𝐹)) = ((2nd𝐺)(⟨𝑋, 𝑌 𝑍)(2nd𝐹)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173  ⟨cop 4131  ⟨cotp 4133  ‘cfv 5804  (class class class)co 6549  2nd c2nd 7058  compcco 15780  Homachoma 16496  compaccoa 16527 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-ot 4134  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-doma 16497  df-coda 16498  df-homa 16499  df-arw 16500  df-coa 16529 This theorem is referenced by:  arwass  16547
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