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Theorem restco 22660
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 3479 . . . . 5 𝑦 ∈ V
21inex1 5317 . . . 4 (𝑦𝐴) ∈ V
3 ineq1 4205 . . . . 5 (𝑥 = (𝑦𝐴) → (𝑥𝐵) = ((𝑦𝐴) ∩ 𝐵))
4 inass 4219 . . . . 5 ((𝑦𝐴) ∩ 𝐵) = (𝑦 ∩ (𝐴𝐵))
53, 4eqtrdi 2789 . . . 4 (𝑥 = (𝑦𝐴) → (𝑥𝐵) = (𝑦 ∩ (𝐴𝐵)))
62, 5abrexco 7240 . . 3 {𝑧 ∣ ∃𝑥 ∈ {𝑤 ∣ ∃𝑦𝐽 𝑤 = (𝑦𝐴)}𝑧 = (𝑥𝐵)} = {𝑧 ∣ ∃𝑦𝐽 𝑧 = (𝑦 ∩ (𝐴𝐵))}
7 eqid 2733 . . . . . 6 (𝑦𝐽 ↦ (𝑦𝐴)) = (𝑦𝐽 ↦ (𝑦𝐴))
87rnmpt 5953 . . . . 5 ran (𝑦𝐽 ↦ (𝑦𝐴)) = {𝑤 ∣ ∃𝑦𝐽 𝑤 = (𝑦𝐴)}
98mpteq1i 5244 . . . 4 (𝑥 ∈ ran (𝑦𝐽 ↦ (𝑦𝐴)) ↦ (𝑥𝐵)) = (𝑥 ∈ {𝑤 ∣ ∃𝑦𝐽 𝑤 = (𝑦𝐴)} ↦ (𝑥𝐵))
109rnmpt 5953 . . 3 ran (𝑥 ∈ ran (𝑦𝐽 ↦ (𝑦𝐴)) ↦ (𝑥𝐵)) = {𝑧 ∣ ∃𝑥 ∈ {𝑤 ∣ ∃𝑦𝐽 𝑤 = (𝑦𝐴)}𝑧 = (𝑥𝐵)}
11 eqid 2733 . . . 4 (𝑦𝐽 ↦ (𝑦 ∩ (𝐴𝐵))) = (𝑦𝐽 ↦ (𝑦 ∩ (𝐴𝐵)))
1211rnmpt 5953 . . 3 ran (𝑦𝐽 ↦ (𝑦 ∩ (𝐴𝐵))) = {𝑧 ∣ ∃𝑦𝐽 𝑧 = (𝑦 ∩ (𝐴𝐵))}
136, 10, 123eqtr4i 2771 . 2 ran (𝑥 ∈ ran (𝑦𝐽 ↦ (𝑦𝐴)) ↦ (𝑥𝐵)) = ran (𝑦𝐽 ↦ (𝑦 ∩ (𝐴𝐵)))
14 restval 17369 . . . . 5 ((𝐽𝑉𝐴𝑊) → (𝐽t 𝐴) = ran (𝑦𝐽 ↦ (𝑦𝐴)))
15143adant3 1133 . . . 4 ((𝐽𝑉𝐴𝑊𝐵𝑋) → (𝐽t 𝐴) = ran (𝑦𝐽 ↦ (𝑦𝐴)))
1615oveq1d 7421 . . 3 ((𝐽𝑉𝐴𝑊𝐵𝑋) → ((𝐽t 𝐴) ↾t 𝐵) = (ran (𝑦𝐽 ↦ (𝑦𝐴)) ↾t 𝐵))
17 ovex 7439 . . . . 5 (𝐽t 𝐴) ∈ V
1815, 17eqeltrrdi 2843 . . . 4 ((𝐽𝑉𝐴𝑊𝐵𝑋) → ran (𝑦𝐽 ↦ (𝑦𝐴)) ∈ V)
19 simp3 1139 . . . 4 ((𝐽𝑉𝐴𝑊𝐵𝑋) → 𝐵𝑋)
20 restval 17369 . . . 4 ((ran (𝑦𝐽 ↦ (𝑦𝐴)) ∈ V ∧ 𝐵𝑋) → (ran (𝑦𝐽 ↦ (𝑦𝐴)) ↾t 𝐵) = ran (𝑥 ∈ ran (𝑦𝐽 ↦ (𝑦𝐴)) ↦ (𝑥𝐵)))
2118, 19, 20syl2anc 585 . . 3 ((𝐽𝑉𝐴𝑊𝐵𝑋) → (ran (𝑦𝐽 ↦ (𝑦𝐴)) ↾t 𝐵) = ran (𝑥 ∈ ran (𝑦𝐽 ↦ (𝑦𝐴)) ↦ (𝑥𝐵)))
2216, 21eqtrd 2773 . 2 ((𝐽𝑉𝐴𝑊𝐵𝑋) → ((𝐽t 𝐴) ↾t 𝐵) = ran (𝑥 ∈ ran (𝑦𝐽 ↦ (𝑦𝐴)) ↦ (𝑥𝐵)))
23 simp1 1137 . . 3 ((𝐽𝑉𝐴𝑊𝐵𝑋) → 𝐽𝑉)
24 inex1g 5319 . . . 4 (𝐴𝑊 → (𝐴𝐵) ∈ V)
25243ad2ant2 1135 . . 3 ((𝐽𝑉𝐴𝑊𝐵𝑋) → (𝐴𝐵) ∈ V)
26 restval 17369 . . 3 ((𝐽𝑉 ∧ (𝐴𝐵) ∈ V) → (𝐽t (𝐴𝐵)) = ran (𝑦𝐽 ↦ (𝑦 ∩ (𝐴𝐵))))
2723, 25, 26syl2anc 585 . 2 ((𝐽𝑉𝐴𝑊𝐵𝑋) → (𝐽t (𝐴𝐵)) = ran (𝑦𝐽 ↦ (𝑦 ∩ (𝐴𝐵))))
2813, 22, 273eqtr4a 2799 1 ((𝐽𝑉𝐴𝑊𝐵𝑋) → ((𝐽t 𝐴) ↾t 𝐵) = (𝐽t (𝐴𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1088   = wceq 1542  wcel 2107  {cab 2710  wrex 3071  Vcvv 3475  cin 3947  cmpt 5231  ran crn 5677  (class class class)co 7406  t crest 17363
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-ov 7409  df-oprab 7410  df-mpo 7411  df-rest 17365
This theorem is referenced by:  restabs  22661  restin  22662  resstopn  22682  ressuss  23759  smfres  45493
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