Theorem homdmcoa 18134

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 2740	. . . 4 ⊢ (Arrow‘𝐶) = (Arrow‘𝐶)
2		homdmcoa.h	. . . 4 ⊢ 𝐻 = (Homa‘𝐶)
3	1, 2	homarw 18113	. . 3 ⊢ (𝑋𝐻𝑌) ⊆ (Arrow‘𝐶)
4		homdmcoa.f	. . 3 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐻𝑌))
5	3, 4	sselid 4006	. 2 ⊢ (𝜑 → 𝐹 ∈ (Arrow‘𝐶))
6	1, 2	homarw 18113	. . 3 ⊢ (𝑌𝐻𝑍) ⊆ (Arrow‘𝐶)
7		homdmcoa.g	. . 3 ⊢ (𝜑 → 𝐺 ∈ (𝑌𝐻𝑍))
8	6, 7	sselid 4006	. 2 ⊢ (𝜑 → 𝐺 ∈ (Arrow‘𝐶))
9	2	homacd 18108	. . . 4 ⊢ (𝐹 ∈ (𝑋𝐻𝑌) → (coda‘𝐹) = 𝑌)
10	4, 9	syl 17	. . 3 ⊢ (𝜑 → (coda‘𝐹) = 𝑌)
11	2	homadm 18107	. . . 4 ⊢ (𝐺 ∈ (𝑌𝐻𝑍) → (doma‘𝐺) = 𝑌)
12	7, 11	syl 17	. . 3 ⊢ (𝜑 → (doma‘𝐺) = 𝑌)
13	10, 12	eqtr4d 2783	. 2 ⊢ (𝜑 → (coda‘𝐹) = (doma‘𝐺))
14		homdmcoa.o	. . 3 ⊢ · = (compa‘𝐶)
15	14, 1	eldmcoa 18132	. 2 ⊢ (𝐺dom · 𝐹 ↔ (𝐹 ∈ (Arrow‘𝐶) ∧ 𝐺 ∈ (Arrow‘𝐶) ∧ (coda‘𝐹) = (doma‘𝐺)))
16	5, 8, 13, 15	syl3anbrc 1343	1 ⊢ (𝜑 → 𝐺dom · 𝐹)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108   class class class wbr 5166  dom cdm 5700  ‘cfv 6573  (class class class)co 7448  dom_acdoma 18087  cod_accoda 18088  Arrowcarw 18089  Hom_achoma 18090  comp_accoa 18121

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-ot 4657  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-doma 18091  df-coda 18092  df-homa 18093  df-arw 18094  df-coa 18123

This theorem is referenced by: (None)