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Theorem coa2 17158
 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 17157 . . 3 (𝜑 → (𝐺 · 𝐹) = ⟨𝑋, 𝑍, ((2nd𝐺)(⟨𝑋, 𝑌 𝑍)(2nd𝐹))⟩)
76fveq2d 6542 . 2 (𝜑 → (2nd ‘(𝐺 · 𝐹)) = (2nd ‘⟨𝑋, 𝑍, ((2nd𝐺)(⟨𝑋, 𝑌 𝑍)(2nd𝐹))⟩))
8 ovex 7048 . . 3 ((2nd𝐺)(⟨𝑋, 𝑌 𝑍)(2nd𝐹)) ∈ V
9 ot3rdg 7561 . . 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 2847 1 (𝜑 → (2nd ‘(𝐺 · 𝐹)) = ((2nd𝐺)(⟨𝑋, 𝑌 𝑍)(2nd𝐹)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1522   ∈ wcel 2081  Vcvv 3437  ⟨cop 4478  ⟨cotp 4480  ‘cfv 6225  (class class class)co 7016  2nd c2nd 7544  compcco 16406  Homachoma 17112  compaccoa 17143 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1777  ax-4 1791  ax-5 1888  ax-6 1947  ax-7 1992  ax-8 2083  ax-9 2091  ax-10 2112  ax-11 2126  ax-12 2141  ax-13 2344  ax-ext 2769  ax-rep 5081  ax-sep 5094  ax-nul 5101  ax-pow 5157  ax-pr 5221  ax-un 7319 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 843  df-3an 1082  df-tru 1525  df-ex 1762  df-nf 1766  df-sb 2043  df-mo 2576  df-eu 2612  df-clab 2776  df-cleq 2788  df-clel 2863  df-nfc 2935  df-ne 2985  df-ral 3110  df-rex 3111  df-reu 3112  df-rab 3114  df-v 3439  df-sbc 3707  df-csb 3812  df-dif 3862  df-un 3864  df-in 3866  df-ss 3874  df-nul 4212  df-if 4382  df-pw 4455  df-sn 4473  df-pr 4475  df-op 4479  df-ot 4481  df-uni 4746  df-iun 4827  df-br 4963  df-opab 5025  df-mpt 5042  df-id 5348  df-xp 5449  df-rel 5450  df-cnv 5451  df-co 5452  df-dm 5453  df-rn 5454  df-res 5455  df-ima 5456  df-iota 6189  df-fun 6227  df-fn 6228  df-f 6229  df-f1 6230  df-fo 6231  df-f1o 6232  df-fv 6233  df-ov 7019  df-oprab 7020  df-mpo 7021  df-1st 7545  df-2nd 7546  df-doma 17113  df-coda 17114  df-homa 17115  df-arw 17116  df-coa 17145 This theorem is referenced by:  arwass  17163
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