Theorem restco 12814

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 2729	. . . . 5 ⊢ 𝑦 ∈ V
2	1	inex1 4116	. . . 4 ⊢ (𝑦 ∩ 𝐴) ∈ V
3		ineq1 3316	. . . . 5 ⊢ (𝑥 = (𝑦 ∩ 𝐴) → (𝑥 ∩ 𝐵) = ((𝑦 ∩ 𝐴) ∩ 𝐵))
4		inass 3332	. . . . 5 ⊢ ((𝑦 ∩ 𝐴) ∩ 𝐵) = (𝑦 ∩ (𝐴 ∩ 𝐵))
5	3, 4	eqtrdi 2215	. . . 4 ⊢ (𝑥 = (𝑦 ∩ 𝐴) → (𝑥 ∩ 𝐵) = (𝑦 ∩ (𝐴 ∩ 𝐵)))
6	2, 5	abrexco 5727	. . 3 ⊢ {𝑧 ∣ ∃𝑥 ∈ {𝑤 ∣ ∃𝑦 ∈ 𝐽 𝑤 = (𝑦 ∩ 𝐴)}𝑧 = (𝑥 ∩ 𝐵)} = {𝑧 ∣ ∃𝑦 ∈ 𝐽 𝑧 = (𝑦 ∩ (𝐴 ∩ 𝐵))}
7		eqid 2165	. . . . . 6 ⊢ (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ 𝐴)) = (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ 𝐴))
8	7	rnmpt 4852	. . . . 5 ⊢ ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ 𝐴)) = {𝑤 ∣ ∃𝑦 ∈ 𝐽 𝑤 = (𝑦 ∩ 𝐴)}
9		mpteq1 4066	. . . . 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 4852	. . 3 ⊢ ran (𝑥 ∈ ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ 𝐴)) ↦ (𝑥 ∩ 𝐵)) = {𝑧 ∣ ∃𝑥 ∈ {𝑤 ∣ ∃𝑦 ∈ 𝐽 𝑤 = (𝑦 ∩ 𝐴)}𝑧 = (𝑥 ∩ 𝐵)}
12		eqid 2165	. . . 4 ⊢ (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ (𝐴 ∩ 𝐵))) = (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ (𝐴 ∩ 𝐵)))
13	12	rnmpt 4852	. . 3 ⊢ ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ (𝐴 ∩ 𝐵))) = {𝑧 ∣ ∃𝑦 ∈ 𝐽 𝑧 = (𝑦 ∩ (𝐴 ∩ 𝐵))}
14	6, 11, 13	3eqtr4i 2196	. 2 ⊢ ran (𝑥 ∈ ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ 𝐴)) ↦ (𝑥 ∩ 𝐵)) = ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ (𝐴 ∩ 𝐵)))
15		restval 12562	. . . . 5 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝐽 ↾t 𝐴) = ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ 𝐴)))
16	15	3adant3 1007	. . . 4 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → (𝐽 ↾t 𝐴) = ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ 𝐴)))
17	16	oveq1d 5857	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → ((𝐽 ↾t 𝐴) ↾t 𝐵) = (ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ 𝐴)) ↾t 𝐵))
18		restfn 12560	. . . . . 6 ⊢ ↾t Fn (V × V)
19		simp1 987	. . . . . . 7 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → 𝐽 ∈ 𝑉)
20	19	elexd 2739	. . . . . 6 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → 𝐽 ∈ V)
21		simp2 988	. . . . . . 7 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → 𝐴 ∈ 𝑊)
22	21	elexd 2739	. . . . . 6 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → 𝐴 ∈ V)
23		fnovex 5875	. . . . . 6 ⊢ (( ↾t Fn (V × V) ∧ 𝐽 ∈ V ∧ 𝐴 ∈ V) → (𝐽 ↾t 𝐴) ∈ V)
24	18, 20, 22, 23	mp3an2i 1332	. . . . 5 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → (𝐽 ↾t 𝐴) ∈ V)
25	16, 24	eqeltrrd 2244	. . . 4 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ 𝐴)) ∈ V)
26		simp3 989	. . . 4 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → 𝐵 ∈ 𝑋)
27		restval 12562	. . . 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 409	. . 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 2198	. 2 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → ((𝐽 ↾t 𝐴) ↾t 𝐵) = ran (𝑥 ∈ ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ 𝐴)) ↦ (𝑥 ∩ 𝐵)))
30		inex1g 4118	. . . 4 ⊢ (𝐴 ∈ 𝑊 → (𝐴 ∩ 𝐵) ∈ V)
31	30	3ad2ant2 1009	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → (𝐴 ∩ 𝐵) ∈ V)
32		restval 12562	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ∈ V) → (𝐽 ↾t (𝐴 ∩ 𝐵)) = ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ (𝐴 ∩ 𝐵))))
33	19, 31, 32	syl2anc 409	. 2 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → (𝐽 ↾t (𝐴 ∩ 𝐵)) = ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ (𝐴 ∩ 𝐵))))
34	14, 29, 33	3eqtr4a 2225	1 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → ((𝐽 ↾t 𝐴) ↾t 𝐵) = (𝐽 ↾t (𝐴 ∩ 𝐵)))

Syntax hints:   → wi 4   ∧ w3a 968   = wceq 1343   ∈ wcel 2136  {cab 2151  ∃wrex 2445  Vcvv 2726   ∩ cin 3115   ↦ cmpt 4043   × cxp 4602  ran crn 4605   Fn wfn 5183  (class class class)co 5842   ↾_t crest 12556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4097  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-csb 3046  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-iun 3868  df-br 3983  df-opab 4044  df-mpt 4045  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-ov 5845  df-oprab 5846  df-mpo 5847  df-1st 6108  df-2nd 6109  df-rest 12558

This theorem is referenced by:  restabs  12815  restin  12816