Theorem restco 13341

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 2740	. . . . 5 ⊢ 𝑦 ∈ V
2	1	inex1 4134	. . . 4 ⊢ (𝑦 ∩ 𝐴) ∈ V
3		ineq1 3329	. . . . 5 ⊢ (𝑥 = (𝑦 ∩ 𝐴) → (𝑥 ∩ 𝐵) = ((𝑦 ∩ 𝐴) ∩ 𝐵))
4		inass 3345	. . . . 5 ⊢ ((𝑦 ∩ 𝐴) ∩ 𝐵) = (𝑦 ∩ (𝐴 ∩ 𝐵))
5	3, 4	eqtrdi 2226	. . . 4 ⊢ (𝑥 = (𝑦 ∩ 𝐴) → (𝑥 ∩ 𝐵) = (𝑦 ∩ (𝐴 ∩ 𝐵)))
6	2, 5	abrexco 5754	. . 3 ⊢ {𝑧 ∣ ∃𝑥 ∈ {𝑤 ∣ ∃𝑦 ∈ 𝐽 𝑤 = (𝑦 ∩ 𝐴)}𝑧 = (𝑥 ∩ 𝐵)} = {𝑧 ∣ ∃𝑦 ∈ 𝐽 𝑧 = (𝑦 ∩ (𝐴 ∩ 𝐵))}
7		eqid 2177	. . . . . 6 ⊢ (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ 𝐴)) = (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ 𝐴))
8	7	rnmpt 4871	. . . . 5 ⊢ ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ 𝐴)) = {𝑤 ∣ ∃𝑦 ∈ 𝐽 𝑤 = (𝑦 ∩ 𝐴)}
9		mpteq1 4084	. . . . 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 4871	. . 3 ⊢ ran (𝑥 ∈ ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ 𝐴)) ↦ (𝑥 ∩ 𝐵)) = {𝑧 ∣ ∃𝑥 ∈ {𝑤 ∣ ∃𝑦 ∈ 𝐽 𝑤 = (𝑦 ∩ 𝐴)}𝑧 = (𝑥 ∩ 𝐵)}
12		eqid 2177	. . . 4 ⊢ (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ (𝐴 ∩ 𝐵))) = (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ (𝐴 ∩ 𝐵)))
13	12	rnmpt 4871	. . 3 ⊢ ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ (𝐴 ∩ 𝐵))) = {𝑧 ∣ ∃𝑦 ∈ 𝐽 𝑧 = (𝑦 ∩ (𝐴 ∩ 𝐵))}
14	6, 11, 13	3eqtr4i 2208	. 2 ⊢ ran (𝑥 ∈ ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ 𝐴)) ↦ (𝑥 ∩ 𝐵)) = ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ (𝐴 ∩ 𝐵)))
15		restval 12642	. . . . 5 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊) → (𝐽 ↾t 𝐴) = ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ 𝐴)))
16	15	3adant3 1017	. . . 4 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → (𝐽 ↾t 𝐴) = ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ 𝐴)))
17	16	oveq1d 5884	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → ((𝐽 ↾t 𝐴) ↾t 𝐵) = (ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ 𝐴)) ↾t 𝐵))
18		restfn 12640	. . . . . 6 ⊢ ↾t Fn (V × V)
19		simp1 997	. . . . . . 7 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → 𝐽 ∈ 𝑉)
20	19	elexd 2750	. . . . . 6 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → 𝐽 ∈ V)
21		simp2 998	. . . . . . 7 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → 𝐴 ∈ 𝑊)
22	21	elexd 2750	. . . . . 6 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → 𝐴 ∈ V)
23		fnovex 5902	. . . . . 6 ⊢ (( ↾t Fn (V × V) ∧ 𝐽 ∈ V ∧ 𝐴 ∈ V) → (𝐽 ↾t 𝐴) ∈ V)
24	18, 20, 22, 23	mp3an2i 1342	. . . . 5 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → (𝐽 ↾t 𝐴) ∈ V)
25	16, 24	eqeltrrd 2255	. . . 4 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ 𝐴)) ∈ V)
26		simp3 999	. . . 4 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → 𝐵 ∈ 𝑋)
27		restval 12642	. . . 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 2210	. 2 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → ((𝐽 ↾t 𝐴) ↾t 𝐵) = ran (𝑥 ∈ ran (𝑦 ∈ 𝐽 ↦ (𝑦 ∩ 𝐴)) ↦ (𝑥 ∩ 𝐵)))
30		inex1g 4136	. . . 4 ⊢ (𝐴 ∈ 𝑊 → (𝐴 ∩ 𝐵) ∈ V)
31	30	3ad2ant2 1019	. . 3 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → (𝐴 ∩ 𝐵) ∈ V)
32		restval 12642	. . 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 2236	1 ⊢ ((𝐽 ∈ 𝑉 ∧ 𝐴 ∈ 𝑊 ∧ 𝐵 ∈ 𝑋) → ((𝐽 ↾t 𝐴) ↾t 𝐵) = (𝐽 ↾t (𝐴 ∩ 𝐵)))

Syntax hints:   → wi 4   ∧ w3a 978   = wceq 1353   ∈ wcel 2148  {cab 2163  ∃wrex 2456  Vcvv 2737   ∩ cin 3128   ↦ cmpt 4061   × cxp 4621  ran crn 4624   Fn wfn 5207  (class class class)co 5869   ↾_t crest 12636

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-id 4290  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-rest 12638

This theorem is referenced by:  restabs  13342  restin  13343