Theorem dmcoass 18241

Description: The domain of composition is a collection of pairs of arrows. (Contributed by Mario Carneiro, 11-Jan-2017.)

Hypotheses

Ref	Expression
coafval.o	⊢ · = (compa‘𝐶)
coafval.a	⊢ 𝐴 = (Arrow‘𝐶)

Assertion

Ref	Expression
dmcoass	⊢ dom · ⊆ (𝐴 × 𝐴)

Proof of Theorem dmcoass

Dummy variables 𝑓 𝑔 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		coafval.o	. . . 4 ⊢ · = (compa‘𝐶)
2		coafval.a	. . . 4 ⊢ 𝐴 = (Arrow‘𝐶)
3		eqid 2760	. . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶)
4	1, 2, 3	coafval 18239	. . 3 ⊢ · = (𝑔 ∈ 𝐴, 𝑓 ∈ {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)} ↦ ⟨(doma‘𝑓), (coda‘𝑔), ((2nd ‘𝑔)(⟨(doma‘𝑓), (doma‘𝑔)⟩(comp‘𝐶)(coda‘𝑔))(2nd ‘𝑓))⟩)
5	4	dmmpossx 8077	. 2 ⊢ dom · ⊆ ∪ 𝑔 ∈ 𝐴 ({𝑔} × {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)})
6		iunss 5002	. . 3 ⊢ (∪ 𝑔 ∈ 𝐴 ({𝑔} × {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)}) ⊆ (𝐴 × 𝐴) ↔ ∀𝑔 ∈ 𝐴 ({𝑔} × {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)}) ⊆ (𝐴 × 𝐴))
7		snssi 4745	. . . 4 ⊢ (𝑔 ∈ 𝐴 → {𝑔} ⊆ 𝐴)
8		ssrab2 4005	. . . 4 ⊢ {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)} ⊆ 𝐴
9		xpss12 5665	. . . 4 ⊢ (({𝑔} ⊆ 𝐴 ∧ {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)} ⊆ 𝐴) → ({𝑔} × {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)}) ⊆ (𝐴 × 𝐴))
10	7, 8, 9	sylancl 598	. . 3 ⊢ (𝑔 ∈ 𝐴 → ({𝑔} × {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)}) ⊆ (𝐴 × 𝐴))
11	6, 10	mprgbir 3083	. 2 ⊢ ∪ 𝑔 ∈ 𝐴 ({𝑔} × {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)}) ⊆ (𝐴 × 𝐴)
12	5, 11	sstri 3917	1 ⊢ dom · ⊆ (𝐴 × 𝐴)

This proof depends on syntax axioms:   = wceq 1569   ∈ wcel 2144  {crab 3412   ⊆ wss 3895  {csn 4583  ⟨cop 4589  ⟨cotp 4591  ∪ ciun 4950   × cxp 5648  dom cdm 5650  ‘cfv 6538  (class class class)co 7420  2^nd c2nd 8000  compcco 17440  dom_acdoma 18195  cod_accoda 18196  Arrowcarw 18197  comp_accoa 18229

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-ot 4592  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-arw 18202  df-coa 18231

This theorem is used by:  coapm  18246