Theorem dmcoass 18133

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 2740	. . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶)
4	1, 2, 3	coafval 18131	. . 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 8107	. 2 ⊢ dom · ⊆ ∪ 𝑔 ∈ 𝐴 ({𝑔} × {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)})
6		iunss 5068	. . 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 4833	. . . 4 ⊢ (𝑔 ∈ 𝐴 → {𝑔} ⊆ 𝐴)
8		ssrab2 4103	. . . 4 ⊢ {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)} ⊆ 𝐴
9		xpss12 5715	. . . 4 ⊢ (({𝑔} ⊆ 𝐴 ∧ {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)} ⊆ 𝐴) → ({𝑔} × {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)}) ⊆ (𝐴 × 𝐴))
10	7, 8, 9	sylancl 585	. . 3 ⊢ (𝑔 ∈ 𝐴 → ({𝑔} × {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)}) ⊆ (𝐴 × 𝐴))
11	6, 10	mprgbir 3074	. 2 ⊢ ∪ 𝑔 ∈ 𝐴 ({𝑔} × {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)}) ⊆ (𝐴 × 𝐴)
12	5, 11	sstri 4018	1 ⊢ dom · ⊆ (𝐴 × 𝐴)

Syntax hints:   = wceq 1537   ∈ wcel 2108  {crab 3443   ⊆ wss 3976  {csn 4648  ⟨cop 4654  ⟨cotp 4656  ∪ ciun 5015   × cxp 5698  dom cdm 5700  ‘cfv 6573  (class class class)co 7448  2^nd c2nd 8029  compcco 17323  dom_acdoma 18087  cod_accoda 18088  Arrowcarw 18089  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-arw 18094  df-coa 18123

This theorem is referenced by:  coapm  18138