Theorem homdmcoa 18080

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 2735	. . . 4 ⊢ (Arrow‘𝐶) = (Arrow‘𝐶)
2		homdmcoa.h	. . . 4 ⊢ 𝐻 = (Homa‘𝐶)
3	1, 2	homarw 18059	. . 3 ⊢ (𝑋𝐻𝑌) ⊆ (Arrow‘𝐶)
4		homdmcoa.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
5	3, 4	sselid 3956	. 2 ⊢ (𝜑 → 𝐹 ∈ (Arrow‘𝐶))
6	1, 2	homarw 18059	. . 3 ⊢ (𝑌𝐻𝑍) ⊆ (Arrow‘𝐶)
7		homdmcoa.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍))
8	6, 7	sselid 3956	. 2 ⊢ (𝜑 → 𝐺 ∈ (Arrow‘𝐶))
9	2	homacd 18054	. . . 4 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (coda‘𝐹) = 𝑌)
10	4, 9	syl 17	. . 3 ⊢ (𝜑 → (coda‘𝐹) = 𝑌)
11	2	homadm 18053	. . . 4 ⊢ (𝐺 ∈ (𝑌𝐻𝑍) → (doma‘𝐺) = 𝑌)
12	7, 11	syl 17	. . 3 ⊢ (𝜑 → (doma‘𝐺) = 𝑌)
13	10, 12	eqtr4d 2773	. 2 ⊢ (𝜑 → (coda‘𝐹) = (doma‘𝐺))
14		homdmcoa.o	. . 3 ⊢ · = (compa‘𝐶)
15	14, 1	eldmcoa 18078	. 2 ⊢ (𝐺dom · 𝐹 ↔ (𝐹 ∈ (Arrow‘𝐶) ∧ 𝐺 ∈ (Arrow‘𝐶) ∧ (coda‘𝐹) = (doma‘𝐺)))
16	5, 8, 13, 15	syl3anbrc 1344	1 ⊢ (𝜑 → 𝐺dom · 𝐹)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2108   class class class wbr 5119  dom cdm 5654  ‘cfv 6531  (class class class)co 7405  dom_acdoma 18033  cod_accoda 18034  Arrowcarw 18035  Hom_achoma 18036  comp_accoa 18067

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-ot 4610  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7988  df-2nd 7989  df-doma 18037  df-coda 18038  df-homa 18039  df-arw 18040  df-coa 18069

This theorem is referenced by: (None)