Theorem restco 21766

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 3498	. . . . 5 ⊢ 𝑦 ∈ V
2	1	inex1 5214	. . . 4 ⊢ (𝑦 ∩ 𝐴) ∈ V
3		ineq1 4181	. . . . 5 ⊢ (𝑥 = (𝑦 ∩ 𝐴) → (𝑥 ∩ 𝐵) = ((𝑦 ∩ 𝐴) ∩ 𝐵))
4		inass 4196	. . . . 5 ⊢ ((𝑦 ∩ 𝐴) ∩ 𝐵) = (𝑦 ∩ (𝐴 ∩ 𝐵))
5	3, 4	syl6eq 2872	. . . 4 ⊢ (𝑥 = (𝑦 ∩ 𝐴) → (𝑥 ∩ 𝐵) = (𝑦 ∩ (𝐴 ∩ 𝐵)))
6	2, 5	abrexco 6997	. . 3 ⊢ {𝑧 ∣ ∃𝑥 ∈ {𝑤 ∣ ∃𝑦 ∈ 𝐽 𝑤 = (𝑦 ∩ 𝐴)}𝑧 = (𝑥 ∩ 𝐵)} = {𝑧 ∣ ∃𝑦 ∈ 𝐽 𝑧 = (𝑦 ∩ (𝐴 ∩ 𝐵))}
7		eqid 2821	. . . . . 6 ⊢ (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ 𝐴)) = (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ 𝐴))
8	7	rnmpt 5822	. . . . 5 ⊢ ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ 𝐴)) = {𝑤 ∣ ∃𝑦 ∈ 𝐽 𝑤 = (𝑦 ∩ 𝐴)}
9	8	mpteq1i 5149	. . . 4 ⊢ (𝑥 ∈ ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ 𝐴)) ↦ (𝑥 ∩ 𝐵)) = (𝑥 ∈ {𝑤 ∣ ∃𝑦 ∈ 𝐽 𝑤 = (𝑦 ∩ 𝐴)} ↦ (𝑥 ∩ 𝐵))
10	9	rnmpt 5822	. . 3 ⊢ ran (𝑥 ∈ ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ 𝐴)) ↦ (𝑥 ∩ 𝐵)) = {𝑧 ∣ ∃𝑥 ∈ {𝑤 ∣ ∃𝑦 ∈ 𝐽 𝑤 = (𝑦 ∩ 𝐴)}𝑧 = (𝑥 ∩ 𝐵)}
11		eqid 2821	. . . 4 ⊢ (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ (𝐴 ∩ 𝐵))) = (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ (𝐴 ∩ 𝐵)))
12	11	rnmpt 5822	. . 3 ⊢ ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ (𝐴 ∩ 𝐵))) = {𝑧 ∣ ∃𝑦 ∈ 𝐽 𝑧 = (𝑦 ∩ (𝐴 ∩ 𝐵))}
13	6, 10, 12	3eqtr4i 2854	. 2 ⊢ ran (𝑥 ∈ ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ 𝐴)) ↦ (𝑥 ∩ 𝐵)) = ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ (𝐴 ∩ 𝐵)))
14		restval 16694	. . . . 5 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝐽 ↾t 𝐴) = ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ 𝐴)))
15	14	3adant3 1128	. . . 4 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → (𝐽 ↾t 𝐴) = ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ 𝐴)))
16	15	oveq1d 7165	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → ((𝐽 ↾t 𝐴) ↾t 𝐵) = (ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ 𝐴)) ↾t 𝐵))
17		ovex 7183	. . . . 5 ⊢ (𝐽 ↾t 𝐴) ∈ V
18	15, 17	eqeltrrdi 2922	. . . 4 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ 𝐴)) ∈ V)
19		simp3 1134	. . . 4 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → 𝐵 ∈ 𝑋)
20		restval 16694	. . . 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)))
21	18, 19, 20	syl2anc 586	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → (ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ 𝐴)) ↾t 𝐵) = ran (𝑥 ∈ ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ 𝐴)) ↦ (𝑥 ∩ 𝐵)))
22	16, 21	eqtrd 2856	. 2 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → ((𝐽 ↾t 𝐴) ↾t 𝐵) = ran (𝑥 ∈ ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ 𝐴)) ↦ (𝑥 ∩ 𝐵)))
23		simp1 1132	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → 𝐽 ∈ 𝑉)
24		inex1g 5216	. . . 4 ⊢ (𝐴 ∈ 𝑊 → (𝐴 ∩ 𝐵) ∈ V)
25	24	3ad2ant2 1130	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → (𝐴 ∩ 𝐵) ∈ V)
26		restval 16694	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ (𝐴 ∩ 𝐵) ∈ V) → (𝐽 ↾t (𝐴 ∩ 𝐵)) = ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ (𝐴 ∩ 𝐵))))
27	23, 25, 26	syl2anc 586	. 2 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → (𝐽 ↾t (𝐴 ∩ 𝐵)) = ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ (𝐴 ∩ 𝐵))))
28	13, 22, 27	3eqtr4a 2882	1 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → ((𝐽 ↾t 𝐴) ↾t 𝐵) = (𝐽 ↾t (𝐴 ∩ 𝐵)))

Syntax hints:   → wi 4   ∧ w3a 1083   = wceq 1533   ∈ wcel 2110  {cab 2799  ∃wrex 3139  Vcvv 3495   ∩ cin 3935   ↦ cmpt 5139  ran crn 5551  (class class class)co 7150   ↾_t crest 16688

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-rep 5183  ax-sep 5196  ax-nul 5203  ax-pr 5322  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rab 3147  df-v 3497  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4833  df-iun 4914  df-br 5060  df-opab 5122  df-mpt 5140  df-id 5455  df-xp 5556  df-rel 5557  df-cnv 5558  df-co 5559  df-dm 5560  df-rn 5561  df-res 5562  df-ima 5563  df-iota 6309  df-fun 6352  df-fn 6353  df-f 6354  df-f1 6355  df-fo 6356  df-f1o 6357  df-fv 6358  df-ov 7153  df-oprab 7154  df-mpo 7155  df-rest 16690

This theorem is referenced by:  restabs  21767  restin  21768  resstopn  21788  ressuss  22866  smfres  43058