Theorem restco 14813

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 2782	. . . . 5 ⊢ 𝑦 ∈ V
2	1	inex1 4197	. . . 4 ⊢ (𝑦 ∩ 𝐴) ∈ V
3		ineq1 3378	. . . . 5 ⊢ (𝑥 = (𝑦 ∩ 𝐴) → (𝑥 ∩ 𝐵) = ((𝑦 ∩ 𝐴) ∩ 𝐵))
4		inass 3394	. . . . 5 ⊢ ((𝑦 ∩ 𝐴) ∩ 𝐵) = (𝑦 ∩ (𝐴 ∩ 𝐵))
5	3, 4	eqtrdi 2258	. . . 4 ⊢ (𝑥 = (𝑦 ∩ 𝐴) → (𝑥 ∩ 𝐵) = (𝑦 ∩ (𝐴 ∩ 𝐵)))
6	2, 5	abrexco 5856	. . 3 ⊢ {𝑧 ∣ ∃𝑥 ∈ {𝑤 ∣ ∃𝑦 ∈ 𝐽 𝑤 = (𝑦 ∩ 𝐴)}𝑧 = (𝑥 ∩ 𝐵)} = {𝑧 ∣ ∃𝑦 ∈ 𝐽 𝑧 = (𝑦 ∩ (𝐴 ∩ 𝐵))}
7		eqid 2209	. . . . . 6 ⊢ (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ 𝐴)) = (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ 𝐴))
8	7	rnmpt 4948	. . . . 5 ⊢ ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ 𝐴)) = {𝑤 ∣ ∃𝑦 ∈ 𝐽 𝑤 = (𝑦 ∩ 𝐴)}
9		mpteq1 4147	. . . . 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 4948	. . 3 ⊢ ran (𝑥 ∈ ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ 𝐴)) ↦ (𝑥 ∩ 𝐵)) = {𝑧 ∣ ∃𝑥 ∈ {𝑤 ∣ ∃𝑦 ∈ 𝐽 𝑤 = (𝑦 ∩ 𝐴)}𝑧 = (𝑥 ∩ 𝐵)}
12		eqid 2209	. . . 4 ⊢ (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ (𝐴 ∩ 𝐵))) = (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ (𝐴 ∩ 𝐵)))
13	12	rnmpt 4948	. . 3 ⊢ ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ (𝐴 ∩ 𝐵))) = {𝑧 ∣ ∃𝑦 ∈ 𝐽 𝑧 = (𝑦 ∩ (𝐴 ∩ 𝐵))}
14	6, 11, 13	3eqtr4i 2240	. 2 ⊢ ran (𝑥 ∈ ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ 𝐴)) ↦ (𝑥 ∩ 𝐵)) = ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ (𝐴 ∩ 𝐵)))
15		restval 13244	. . . . 5 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝐽 ↾t 𝐴) = ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ 𝐴)))
16	15	3adant3 1022	. . . 4 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → (𝐽 ↾t 𝐴) = ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ 𝐴)))
17	16	oveq1d 5989	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → ((𝐽 ↾t 𝐴) ↾t 𝐵) = (ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ 𝐴)) ↾t 𝐵))
18		restfn 13242	. . . . . 6 ⊢ ↾t Fn (V × V)
19		simp1 1002	. . . . . . 7 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → 𝐽 ∈ 𝑉)
20	19	elexd 2793	. . . . . 6 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → 𝐽 ∈ V)
21		simp2 1003	. . . . . . 7 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → 𝐴 ∈ 𝑊)
22	21	elexd 2793	. . . . . 6 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → 𝐴 ∈ V)
23		fnovex 6007	. . . . . 6 ⊢ (( ↾t Fn (V × V) ∧ 𝐽 ∈ V ∧ 𝐴 ∈ V) → (𝐽 ↾t 𝐴) ∈ V)
24	18, 20, 22, 23	mp3an2i 1357	. . . . 5 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → (𝐽 ↾t 𝐴) ∈ V)
25	16, 24	eqeltrrd 2287	. . . 4 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ 𝐴)) ∈ V)
26		simp3 1004	. . . 4 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → 𝐵 ∈ 𝑋)
27		restval 13244	. . . 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 2242	. 2 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → ((𝐽 ↾t 𝐴) ↾t 𝐵) = ran (𝑥 ∈ ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ 𝐴)) ↦ (𝑥 ∩ 𝐵)))
30		inex1g 4199	. . . 4 ⊢ (𝐴 ∈ 𝑊 → (𝐴 ∩ 𝐵) ∈ V)
31	30	3ad2ant2 1024	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → (𝐴 ∩ 𝐵) ∈ V)
32		restval 13244	. . 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 2268	1 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → ((𝐽 ↾t 𝐴) ↾t 𝐵) = (𝐽 ↾t (𝐴 ∩ 𝐵)))

Syntax hints:   → wi 4   ∧ w3a 983   = wceq 1375   ∈ wcel 2180  {cab 2195  ∃wrex 2489  Vcvv 2779   ∩ cin 3176   ↦ cmpt 4124   × cxp 4694  ran crn 4697   Fn wfn 5289  (class class class)co 5974   ↾_t crest 13238

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 617  ax-in2 618  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-coll 4178  ax-sep 4181  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-setind 4606

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-ral 2493  df-rex 2494  df-reu 2495  df-rab 2497  df-v 2781  df-sbc 3009  df-csb 3105  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-iun 3946  df-br 4063  df-opab 4125  df-mpt 4126  df-id 4361  df-xp 4702  df-rel 4703  df-cnv 4704  df-co 4705  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709  df-iota 5254  df-fun 5296  df-fn 5297  df-f 5298  df-f1 5299  df-fo 5300  df-f1o 5301  df-fv 5302  df-ov 5977  df-oprab 5978  df-mpo 5979  df-1st 6256  df-2nd 6257  df-rest 13240

This theorem is referenced by:  restabs  14814  restin  14815