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Theorem restco 21190
Description: Composition of subspaces. (Contributed by Mario Carneiro, 15-Dec-2013.) (Revised by Mario Carneiro, 1-May-2015.)
Assertion
Ref Expression
restco ((𝐽𝑉𝐴𝑊𝐵𝑋) → ((𝐽t 𝐴) ↾t 𝐵) = (𝐽t (𝐴𝐵)))

Proof of Theorem restco
Dummy variables 𝑥 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3343 . . . . 5 𝑦 ∈ V
21inex1 4951 . . . 4 (𝑦𝐴) ∈ V
3 ineq1 3950 . . . . 5 (𝑥 = (𝑦𝐴) → (𝑥𝐵) = ((𝑦𝐴) ∩ 𝐵))
4 inass 3966 . . . . 5 ((𝑦𝐴) ∩ 𝐵) = (𝑦 ∩ (𝐴𝐵))
53, 4syl6eq 2810 . . . 4 (𝑥 = (𝑦𝐴) → (𝑥𝐵) = (𝑦 ∩ (𝐴𝐵)))
62, 5abrexco 6666 . . 3 {𝑧 ∣ ∃𝑥 ∈ {𝑤 ∣ ∃𝑦𝐽 𝑤 = (𝑦𝐴)}𝑧 = (𝑥𝐵)} = {𝑧 ∣ ∃𝑦𝐽 𝑧 = (𝑦 ∩ (𝐴𝐵))}
7 eqid 2760 . . . . . 6 (𝑦𝐽 ↦ (𝑦𝐴)) = (𝑦𝐽 ↦ (𝑦𝐴))
87rnmpt 5526 . . . . 5 ran (𝑦𝐽 ↦ (𝑦𝐴)) = {𝑤 ∣ ∃𝑦𝐽 𝑤 = (𝑦𝐴)}
9 mpteq1 4889 . . . . 5 (ran (𝑦𝐽 ↦ (𝑦𝐴)) = {𝑤 ∣ ∃𝑦𝐽 𝑤 = (𝑦𝐴)} → (𝑥 ∈ ran (𝑦𝐽 ↦ (𝑦𝐴)) ↦ (𝑥𝐵)) = (𝑥 ∈ {𝑤 ∣ ∃𝑦𝐽 𝑤 = (𝑦𝐴)} ↦ (𝑥𝐵)))
108, 9ax-mp 5 . . . 4 (𝑥 ∈ ran (𝑦𝐽 ↦ (𝑦𝐴)) ↦ (𝑥𝐵)) = (𝑥 ∈ {𝑤 ∣ ∃𝑦𝐽 𝑤 = (𝑦𝐴)} ↦ (𝑥𝐵))
1110rnmpt 5526 . . 3 ran (𝑥 ∈ ran (𝑦𝐽 ↦ (𝑦𝐴)) ↦ (𝑥𝐵)) = {𝑧 ∣ ∃𝑥 ∈ {𝑤 ∣ ∃𝑦𝐽 𝑤 = (𝑦𝐴)}𝑧 = (𝑥𝐵)}
12 eqid 2760 . . . 4 (𝑦𝐽 ↦ (𝑦 ∩ (𝐴𝐵))) = (𝑦𝐽 ↦ (𝑦 ∩ (𝐴𝐵)))
1312rnmpt 5526 . . 3 ran (𝑦𝐽 ↦ (𝑦 ∩ (𝐴𝐵))) = {𝑧 ∣ ∃𝑦𝐽 𝑧 = (𝑦 ∩ (𝐴𝐵))}
146, 11, 133eqtr4i 2792 . 2 ran (𝑥 ∈ ran (𝑦𝐽 ↦ (𝑦𝐴)) ↦ (𝑥𝐵)) = ran (𝑦𝐽 ↦ (𝑦 ∩ (𝐴𝐵)))
15 restval 16309 . . . . 5 ((𝐽𝑉𝐴𝑊) → (𝐽t 𝐴) = ran (𝑦𝐽 ↦ (𝑦𝐴)))
16153adant3 1127 . . . 4 ((𝐽𝑉𝐴𝑊𝐵𝑋) → (𝐽t 𝐴) = ran (𝑦𝐽 ↦ (𝑦𝐴)))
1716oveq1d 6829 . . 3 ((𝐽𝑉𝐴𝑊𝐵𝑋) → ((𝐽t 𝐴) ↾t 𝐵) = (ran (𝑦𝐽 ↦ (𝑦𝐴)) ↾t 𝐵))
18 ovex 6842 . . . . 5 (𝐽t 𝐴) ∈ V
1916, 18syl6eqelr 2848 . . . 4 ((𝐽𝑉𝐴𝑊𝐵𝑋) → ran (𝑦𝐽 ↦ (𝑦𝐴)) ∈ V)
20 simp3 1133 . . . 4 ((𝐽𝑉𝐴𝑊𝐵𝑋) → 𝐵𝑋)
21 restval 16309 . . . 4 ((ran (𝑦𝐽 ↦ (𝑦𝐴)) ∈ V ∧ 𝐵𝑋) → (ran (𝑦𝐽 ↦ (𝑦𝐴)) ↾t 𝐵) = ran (𝑥 ∈ ran (𝑦𝐽 ↦ (𝑦𝐴)) ↦ (𝑥𝐵)))
2219, 20, 21syl2anc 696 . . 3 ((𝐽𝑉𝐴𝑊𝐵𝑋) → (ran (𝑦𝐽 ↦ (𝑦𝐴)) ↾t 𝐵) = ran (𝑥 ∈ ran (𝑦𝐽 ↦ (𝑦𝐴)) ↦ (𝑥𝐵)))
2317, 22eqtrd 2794 . 2 ((𝐽𝑉𝐴𝑊𝐵𝑋) → ((𝐽t 𝐴) ↾t 𝐵) = ran (𝑥 ∈ ran (𝑦𝐽 ↦ (𝑦𝐴)) ↦ (𝑥𝐵)))
24 simp1 1131 . . 3 ((𝐽𝑉𝐴𝑊𝐵𝑋) → 𝐽𝑉)
25 inex1g 4953 . . . 4 (𝐴𝑊 → (𝐴𝐵) ∈ V)
26253ad2ant2 1129 . . 3 ((𝐽𝑉𝐴𝑊𝐵𝑋) → (𝐴𝐵) ∈ V)
27 restval 16309 . . 3 ((𝐽𝑉 ∧ (𝐴𝐵) ∈ V) → (𝐽t (𝐴𝐵)) = ran (𝑦𝐽 ↦ (𝑦 ∩ (𝐴𝐵))))
2824, 26, 27syl2anc 696 . 2 ((𝐽𝑉𝐴𝑊𝐵𝑋) → (𝐽t (𝐴𝐵)) = ran (𝑦𝐽 ↦ (𝑦 ∩ (𝐴𝐵))))
2914, 23, 283eqtr4a 2820 1 ((𝐽𝑉𝐴𝑊𝐵𝑋) → ((𝐽t 𝐴) ↾t 𝐵) = (𝐽t (𝐴𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1072   = wceq 1632  wcel 2139  {cab 2746  wrex 3051  Vcvv 3340  cin 3714  cmpt 4881  ran crn 5267  (class class class)co 6814  t crest 16303
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-un 7115
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-rest 16305
This theorem is referenced by:  restabs  21191  restin  21192  resstopn  21212  ressuss  22288  smfres  41521
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