Theorem dmcoass 16716

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 2622	. . . 4 ⊢ (comp‘𝐶) = (comp‘𝐶)
4	1, 2, 3	coafval 16714	. . 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	dmmpt2ssx 7235	. 2 ⊢ dom · ⊆ ∪ 𝑔 ∈ 𝐴 ({𝑔} × {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)})
6		iunss 4561	. . 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 4339	. . . 4 ⊢ (𝑔 ∈ 𝐴 → {𝑔} ⊆ 𝐴)
8		ssrab2 3687	. . . 4 ⊢ {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)} ⊆ 𝐴
9		xpss12 5225	. . . 4 ⊢ (({𝑔} ⊆ 𝐴 ∧ {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)} ⊆ 𝐴) → ({𝑔} × {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)}) ⊆ (𝐴 × 𝐴))
10	7, 8, 9	sylancl 694	. . 3 ⊢ (𝑔 ∈ 𝐴 → ({𝑔} × {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)}) ⊆ (𝐴 × 𝐴))
11	6, 10	mprgbir 2927	. 2 ⊢ ∪ 𝑔 ∈ 𝐴 ({𝑔} × {ℎ ∈ 𝐴 ∣ (coda‘ℎ) = (doma‘𝑔)}) ⊆ (𝐴 × 𝐴)
12	5, 11	sstri 3612	1 ⊢ dom · ⊆ (𝐴 × 𝐴)

Syntax hints:   = wceq 1483   ∈ wcel 1990  {crab 2916   ⊆ wss 3574  {csn 4177  ⟨cop 4183  ⟨cotp 4185  ∪ ciun 4520   × cxp 5112  dom cdm 5114  ‘cfv 5888  (class class class)co 6650  2^nd c2nd 7167  compcco 15953  dom_acdoma 16670  cod_accoda 16671  Arrowcarw 16672  comp_accoa 16704

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-ot 4186  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-arw 16677  df-coa 16706

This theorem is referenced by:  coapm  16721