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Theorem restco 21767
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 3472 . . . . 5 𝑦 ∈ V
21inex1 5197 . . . 4 (𝑦𝐴) ∈ V
3 ineq1 4155 . . . . 5 (𝑥 = (𝑦𝐴) → (𝑥𝐵) = ((𝑦𝐴) ∩ 𝐵))
4 inass 4170 . . . . 5 ((𝑦𝐴) ∩ 𝐵) = (𝑦 ∩ (𝐴𝐵))
53, 4syl6eq 2873 . . . 4 (𝑥 = (𝑦𝐴) → (𝑥𝐵) = (𝑦 ∩ (𝐴𝐵)))
62, 5abrexco 6986 . . 3 {𝑧 ∣ ∃𝑥 ∈ {𝑤 ∣ ∃𝑦𝐽 𝑤 = (𝑦𝐴)}𝑧 = (𝑥𝐵)} = {𝑧 ∣ ∃𝑦𝐽 𝑧 = (𝑦 ∩ (𝐴𝐵))}
7 eqid 2822 . . . . . 6 (𝑦𝐽 ↦ (𝑦𝐴)) = (𝑦𝐽 ↦ (𝑦𝐴))
87rnmpt 5804 . . . . 5 ran (𝑦𝐽 ↦ (𝑦𝐴)) = {𝑤 ∣ ∃𝑦𝐽 𝑤 = (𝑦𝐴)}
98mpteq1i 5132 . . . 4 (𝑥 ∈ ran (𝑦𝐽 ↦ (𝑦𝐴)) ↦ (𝑥𝐵)) = (𝑥 ∈ {𝑤 ∣ ∃𝑦𝐽 𝑤 = (𝑦𝐴)} ↦ (𝑥𝐵))
109rnmpt 5804 . . 3 ran (𝑥 ∈ ran (𝑦𝐽 ↦ (𝑦𝐴)) ↦ (𝑥𝐵)) = {𝑧 ∣ ∃𝑥 ∈ {𝑤 ∣ ∃𝑦𝐽 𝑤 = (𝑦𝐴)}𝑧 = (𝑥𝐵)}
11 eqid 2822 . . . 4 (𝑦𝐽 ↦ (𝑦 ∩ (𝐴𝐵))) = (𝑦𝐽 ↦ (𝑦 ∩ (𝐴𝐵)))
1211rnmpt 5804 . . 3 ran (𝑦𝐽 ↦ (𝑦 ∩ (𝐴𝐵))) = {𝑧 ∣ ∃𝑦𝐽 𝑧 = (𝑦 ∩ (𝐴𝐵))}
136, 10, 123eqtr4i 2855 . 2 ran (𝑥 ∈ ran (𝑦𝐽 ↦ (𝑦𝐴)) ↦ (𝑥𝐵)) = ran (𝑦𝐽 ↦ (𝑦 ∩ (𝐴𝐵)))
14 restval 16691 . . . . 5 ((𝐽𝑉𝐴𝑊) → (𝐽t 𝐴) = ran (𝑦𝐽 ↦ (𝑦𝐴)))
15143adant3 1129 . . . 4 ((𝐽𝑉𝐴𝑊𝐵𝑋) → (𝐽t 𝐴) = ran (𝑦𝐽 ↦ (𝑦𝐴)))
1615oveq1d 7155 . . 3 ((𝐽𝑉𝐴𝑊𝐵𝑋) → ((𝐽t 𝐴) ↾t 𝐵) = (ran (𝑦𝐽 ↦ (𝑦𝐴)) ↾t 𝐵))
17 ovex 7173 . . . . 5 (𝐽t 𝐴) ∈ V
1815, 17eqeltrrdi 2923 . . . 4 ((𝐽𝑉𝐴𝑊𝐵𝑋) → ran (𝑦𝐽 ↦ (𝑦𝐴)) ∈ V)
19 simp3 1135 . . . 4 ((𝐽𝑉𝐴𝑊𝐵𝑋) → 𝐵𝑋)
20 restval 16691 . . . 4 ((ran (𝑦𝐽 ↦ (𝑦𝐴)) ∈ V ∧ 𝐵𝑋) → (ran (𝑦𝐽 ↦ (𝑦𝐴)) ↾t 𝐵) = ran (𝑥 ∈ ran (𝑦𝐽 ↦ (𝑦𝐴)) ↦ (𝑥𝐵)))
2118, 19, 20syl2anc 587 . . 3 ((𝐽𝑉𝐴𝑊𝐵𝑋) → (ran (𝑦𝐽 ↦ (𝑦𝐴)) ↾t 𝐵) = ran (𝑥 ∈ ran (𝑦𝐽 ↦ (𝑦𝐴)) ↦ (𝑥𝐵)))
2216, 21eqtrd 2857 . 2 ((𝐽𝑉𝐴𝑊𝐵𝑋) → ((𝐽t 𝐴) ↾t 𝐵) = ran (𝑥 ∈ ran (𝑦𝐽 ↦ (𝑦𝐴)) ↦ (𝑥𝐵)))
23 simp1 1133 . . 3 ((𝐽𝑉𝐴𝑊𝐵𝑋) → 𝐽𝑉)
24 inex1g 5199 . . . 4 (𝐴𝑊 → (𝐴𝐵) ∈ V)
25243ad2ant2 1131 . . 3 ((𝐽𝑉𝐴𝑊𝐵𝑋) → (𝐴𝐵) ∈ V)
26 restval 16691 . . 3 ((𝐽𝑉 ∧ (𝐴𝐵) ∈ V) → (𝐽t (𝐴𝐵)) = ran (𝑦𝐽 ↦ (𝑦 ∩ (𝐴𝐵))))
2723, 25, 26syl2anc 587 . 2 ((𝐽𝑉𝐴𝑊𝐵𝑋) → (𝐽t (𝐴𝐵)) = ran (𝑦𝐽 ↦ (𝑦 ∩ (𝐴𝐵))))
2813, 22, 273eqtr4a 2883 1 ((𝐽𝑉𝐴𝑊𝐵𝑋) → ((𝐽t 𝐴) ↾t 𝐵) = (𝐽t (𝐴𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1084   = wceq 1538  wcel 2114  {cab 2800  wrex 3131  Vcvv 3469  cin 3907  cmpt 5122  ran crn 5533  (class class class)co 7140  t crest 16685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-rep 5166  ax-sep 5179  ax-nul 5186  ax-pr 5307  ax-un 7446
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ne 3012  df-ral 3135  df-rex 3136  df-reu 3137  df-rab 3139  df-v 3471  df-sbc 3748  df-csb 3856  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4814  df-iun 4896  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5437  df-xp 5538  df-rel 5539  df-cnv 5540  df-co 5541  df-dm 5542  df-rn 5543  df-res 5544  df-ima 5545  df-iota 6293  df-fun 6336  df-fn 6337  df-f 6338  df-f1 6339  df-fo 6340  df-f1o 6341  df-fv 6342  df-ov 7143  df-oprab 7144  df-mpo 7145  df-rest 16687
This theorem is referenced by:  restabs  21768  restin  21769  resstopn  21789  ressuss  22867  smfres  43362
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