Theorem homdmcoa 16540

Description: If 𝐹:𝑋⟶𝑌 and 𝐺:𝑌⟶𝑍, then 𝐺 and 𝐹 are composable. (Contributed by Mario Carneiro, 11-Jan-2017.)

Hypotheses

Ref	Expression
homdmcoa.o	⊢ · = (compa‘𝐶)
homdmcoa.h	⊢ 𝐻 = (Homa‘𝐶)
homdmcoa.f	⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
homdmcoa.g	⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍))

Assertion

Ref	Expression
homdmcoa	⊢ (𝜑 → 𝐺dom · 𝐹)

Proof of Theorem homdmcoa

Step	Hyp	Ref	Expression
1		eqid 2610	. . . 4 ⊢ (Arrow‘𝐶) = (Arrow‘𝐶)
2		homdmcoa.h	. . . 4 ⊢ 𝐻 = (Homa‘𝐶)
3	1, 2	homarw 16519	. . 3 ⊢ (𝑋𝐻𝑌) ⊆ (Arrow‘𝐶)
4		homdmcoa.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
5	3, 4	sseldi 3566	. 2 ⊢ (𝜑 → 𝐹 ∈ (Arrow‘𝐶))
6	1, 2	homarw 16519	. . 3 ⊢ (𝑌𝐻𝑍) ⊆ (Arrow‘𝐶)
7		homdmcoa.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍))
8	6, 7	sseldi 3566	. 2 ⊢ (𝜑 → 𝐺 ∈ (Arrow‘𝐶))
9	2	homacd 16514	. . . 4 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (coda‘𝐹) = 𝑌)
10	4, 9	syl 17	. . 3 ⊢ (𝜑 → (coda‘𝐹) = 𝑌)
11	2	homadm 16513	. . . 4 ⊢ (𝐺 ∈ (𝑌𝐻𝑍) → (doma‘𝐺) = 𝑌)
12	7, 11	syl 17	. . 3 ⊢ (𝜑 → (doma‘𝐺) = 𝑌)
13	10, 12	eqtr4d 2647	. 2 ⊢ (𝜑 → (coda‘𝐹) = (doma‘𝐺))
14		homdmcoa.o	. . 3 ⊢ · = (compa‘𝐶)
15	14, 1	eldmcoa 16538	. 2 ⊢ (𝐺dom · 𝐹 ↔ (𝐹 ∈ (Arrow‘𝐶) ∧ 𝐺 ∈ (Arrow‘𝐶) ∧ (coda‘𝐹) = (doma‘𝐺)))
16	5, 8, 13, 15	syl3anbrc 1239	1 ⊢ (𝜑 → 𝐺dom · 𝐹)

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977   class class class wbr 4583  dom cdm 5038  ‘cfv 5804  (class class class)co 6549  dom_acdoma 16493  cod_accoda 16494  Arrowcarw 16495  Hom_achoma 16496  comp_accoa 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: (None)