Theorem restco 14353

Description: Composition of subspaces. (Contributed by Mario Carneiro, 15-Dec-2013.) (Revised by Mario Carneiro, 1-May-2015.)

Assertion

Ref	Expression
restco	⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → ((𝐽 ↾t 𝐴) ↾t 𝐵) = (𝐽 ↾t (𝐴 ∩ 𝐵)))

Proof of Theorem restco

Dummy variables 𝑥 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2763	. . . . 5 ⊢ 𝑦 ∈ V
2	1	inex1 4164	. . . 4 ⊢ (𝑦 ∩ 𝐴) ∈ V
3		ineq1 3354	. . . . 5 ⊢ (𝑥 = (𝑦 ∩ 𝐴) → (𝑥 ∩ 𝐵) = ((𝑦 ∩ 𝐴) ∩ 𝐵))
4		inass 3370	. . . . 5 ⊢ ((𝑦 ∩ 𝐴) ∩ 𝐵) = (𝑦 ∩ (𝐴 ∩ 𝐵))
5	3, 4	eqtrdi 2242	. . . 4 ⊢ (𝑥 = (𝑦 ∩ 𝐴) → (𝑥 ∩ 𝐵) = (𝑦 ∩ (𝐴 ∩ 𝐵)))
6	2, 5	abrexco 5803	. . 3 ⊢ {𝑧 ∣ ∃𝑥 ∈ {𝑤 ∣ ∃𝑦 ∈ 𝐽 𝑤 = (𝑦 ∩ 𝐴)}𝑧 = (𝑥 ∩ 𝐵)} = {𝑧 ∣ ∃𝑦 ∈ 𝐽 𝑧 = (𝑦 ∩ (𝐴 ∩ 𝐵))}
7		eqid 2193	. . . . . 6 ⊢ (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ 𝐴)) = (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ 𝐴))
8	7	rnmpt 4911	. . . . 5 ⊢ ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ 𝐴)) = {𝑤 ∣ ∃𝑦 ∈ 𝐽 𝑤 = (𝑦 ∩ 𝐴)}
9		mpteq1 4114	. . . . 5 ⊢ (ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ 𝐴)) = {𝑤 ∣ ∃𝑦 ∈ 𝐽 𝑤 = (𝑦 ∩ 𝐴)} → (𝑥 ∈ ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ 𝐴)) ↦ (𝑥 ∩ 𝐵)) = (𝑥 ∈ {𝑤 ∣ ∃𝑦 ∈ 𝐽 𝑤 = (𝑦 ∩ 𝐴)} ↦ (𝑥 ∩ 𝐵)))
10	8, 9	ax-mp 5	. . . 4 ⊢ (𝑥 ∈ ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ 𝐴)) ↦ (𝑥 ∩ 𝐵)) = (𝑥 ∈ {𝑤 ∣ ∃𝑦 ∈ 𝐽 𝑤 = (𝑦 ∩ 𝐴)} ↦ (𝑥 ∩ 𝐵))
11	10	rnmpt 4911	. . 3 ⊢ ran (𝑥 ∈ ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ 𝐴)) ↦ (𝑥 ∩ 𝐵)) = {𝑧 ∣ ∃𝑥 ∈ {𝑤 ∣ ∃𝑦 ∈ 𝐽 𝑤 = (𝑦 ∩ 𝐴)}𝑧 = (𝑥 ∩ 𝐵)}
12		eqid 2193	. . . 4 ⊢ (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ (𝐴 ∩ 𝐵))) = (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ (𝐴 ∩ 𝐵)))
13	12	rnmpt 4911	. . 3 ⊢ ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ (𝐴 ∩ 𝐵))) = {𝑧 ∣ ∃𝑦 ∈ 𝐽 𝑧 = (𝑦 ∩ (𝐴 ∩ 𝐵))}
14	6, 11, 13	3eqtr4i 2224	. 2 ⊢ ran (𝑥 ∈ ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ 𝐴)) ↦ (𝑥 ∩ 𝐵)) = ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ (𝐴 ∩ 𝐵)))
15		restval 12859	. . . . 5 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝐽 ↾t 𝐴) = ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ 𝐴)))
16	15	3adant3 1019	. . . 4 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → (𝐽 ↾t 𝐴) = ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ 𝐴)))
17	16	oveq1d 5934	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → ((𝐽 ↾t 𝐴) ↾t 𝐵) = (ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ 𝐴)) ↾t 𝐵))
18		restfn 12857	. . . . . 6 ⊢ ↾t Fn (V × V)
19		simp1 999	. . . . . . 7 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → 𝐽 ∈ 𝑉)
20	19	elexd 2773	. . . . . 6 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → 𝐽 ∈ V)
21		simp2 1000	. . . . . . 7 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → 𝐴 ∈ 𝑊)
22	21	elexd 2773	. . . . . 6 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → 𝐴 ∈ V)
23		fnovex 5952	. . . . . 6 ⊢ (( ↾t Fn (V × V) ∧ 𝐽 ∈ V ∧ 𝐴 ∈ V) → (𝐽 ↾t 𝐴) ∈ V)
24	18, 20, 22, 23	mp3an2i 1353	. . . . 5 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → (𝐽 ↾t 𝐴) ∈ V)
25	16, 24	eqeltrrd 2271	. . . 4 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ 𝐴)) ∈ V)
26		simp3 1001	. . . 4 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → 𝐵 ∈ 𝑋)
27		restval 12859	. . . 4 ⊢ ((ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ 𝐴)) ∈ V ∧ 𝐵 ∈ 𝑋) → (ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ 𝐴)) ↾t 𝐵) = ran (𝑥 ∈ ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ 𝐴)) ↦ (𝑥 ∩ 𝐵)))
28	25, 26, 27	syl2anc 411	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → (ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ 𝐴)) ↾t 𝐵) = ran (𝑥 ∈ ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ 𝐴)) ↦ (𝑥 ∩ 𝐵)))
29	17, 28	eqtrd 2226	. 2 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → ((𝐽 ↾t 𝐴) ↾t 𝐵) = ran (𝑥 ∈ ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ 𝐴)) ↦ (𝑥 ∩ 𝐵)))
30		inex1g 4166	. . . 4 ⊢ (𝐴 ∈ 𝑊 → (𝐴 ∩ 𝐵) ∈ V)
31	30	3ad2ant2 1021	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → (𝐴 ∩ 𝐵) ∈ V)
32		restval 12859	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ∈ V) → (𝐽 ↾t (𝐴 ∩ 𝐵)) = ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ (𝐴 ∩ 𝐵))))
33	19, 31, 32	syl2anc 411	. 2 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → (𝐽 ↾t (𝐴 ∩ 𝐵)) = ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ (𝐴 ∩ 𝐵))))
34	14, 29, 33	3eqtr4a 2252	1 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → ((𝐽 ↾t 𝐴) ↾t 𝐵) = (𝐽 ↾t (𝐴 ∩ 𝐵)))

Syntax hints:   → wi 4   ∧ w3a 980   = wceq 1364   ∈ wcel 2164  {cab 2179  ∃wrex 2473  Vcvv 2760   ∩ cin 3153   ↦ cmpt 4091   × cxp 4658  ran crn 4661   Fn wfn 5250  (class class class)co 5919   ↾_t crest 12853

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4145  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2987  df-csb 3082  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-iun 3915  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5922  df-oprab 5923  df-mpo 5924  df-1st 6195  df-2nd 6196  df-rest 12855

This theorem is referenced by:  restabs  14354  restin  14355