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Theorem restco 23172
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 3484 . . . . 5 𝑦 ∈ V
21inex1 5317 . . . 4 (𝑦𝐴) ∈ V
3 ineq1 4213 . . . . 5 (𝑥 = (𝑦𝐴) → (𝑥𝐵) = ((𝑦𝐴) ∩ 𝐵))
4 inass 4228 . . . . 5 ((𝑦𝐴) ∩ 𝐵) = (𝑦 ∩ (𝐴𝐵))
53, 4eqtrdi 2793 . . . 4 (𝑥 = (𝑦𝐴) → (𝑥𝐵) = (𝑦 ∩ (𝐴𝐵)))
62, 5abrexco 7264 . . 3 {𝑧 ∣ ∃𝑥 ∈ {𝑤 ∣ ∃𝑦𝐽 𝑤 = (𝑦𝐴)}𝑧 = (𝑥𝐵)} = {𝑧 ∣ ∃𝑦𝐽 𝑧 = (𝑦 ∩ (𝐴𝐵))}
7 eqid 2737 . . . . . 6 (𝑦𝐽 ↦ (𝑦𝐴)) = (𝑦𝐽 ↦ (𝑦𝐴))
87rnmpt 5968 . . . . 5 ran (𝑦𝐽 ↦ (𝑦𝐴)) = {𝑤 ∣ ∃𝑦𝐽 𝑤 = (𝑦𝐴)}
98mpteq1i 5238 . . . 4 (𝑥 ∈ ran (𝑦𝐽 ↦ (𝑦𝐴)) ↦ (𝑥𝐵)) = (𝑥 ∈ {𝑤 ∣ ∃𝑦𝐽 𝑤 = (𝑦𝐴)} ↦ (𝑥𝐵))
109rnmpt 5968 . . 3 ran (𝑥 ∈ ran (𝑦𝐽 ↦ (𝑦𝐴)) ↦ (𝑥𝐵)) = {𝑧 ∣ ∃𝑥 ∈ {𝑤 ∣ ∃𝑦𝐽 𝑤 = (𝑦𝐴)}𝑧 = (𝑥𝐵)}
11 eqid 2737 . . . 4 (𝑦𝐽 ↦ (𝑦 ∩ (𝐴𝐵))) = (𝑦𝐽 ↦ (𝑦 ∩ (𝐴𝐵)))
1211rnmpt 5968 . . 3 ran (𝑦𝐽 ↦ (𝑦 ∩ (𝐴𝐵))) = {𝑧 ∣ ∃𝑦𝐽 𝑧 = (𝑦 ∩ (𝐴𝐵))}
136, 10, 123eqtr4i 2775 . 2 ran (𝑥 ∈ ran (𝑦𝐽 ↦ (𝑦𝐴)) ↦ (𝑥𝐵)) = ran (𝑦𝐽 ↦ (𝑦 ∩ (𝐴𝐵)))
14 restval 17471 . . . . 5 ((𝐽𝑉𝐴𝑊) → (𝐽t 𝐴) = ran (𝑦𝐽 ↦ (𝑦𝐴)))
15143adant3 1133 . . . 4 ((𝐽𝑉𝐴𝑊𝐵𝑋) → (𝐽t 𝐴) = ran (𝑦𝐽 ↦ (𝑦𝐴)))
1615oveq1d 7446 . . 3 ((𝐽𝑉𝐴𝑊𝐵𝑋) → ((𝐽t 𝐴) ↾t 𝐵) = (ran (𝑦𝐽 ↦ (𝑦𝐴)) ↾t 𝐵))
17 ovex 7464 . . . . 5 (𝐽t 𝐴) ∈ V
1815, 17eqeltrrdi 2850 . . . 4 ((𝐽𝑉𝐴𝑊𝐵𝑋) → ran (𝑦𝐽 ↦ (𝑦𝐴)) ∈ V)
19 simp3 1139 . . . 4 ((𝐽𝑉𝐴𝑊𝐵𝑋) → 𝐵𝑋)
20 restval 17471 . . . 4 ((ran (𝑦𝐽 ↦ (𝑦𝐴)) ∈ V ∧ 𝐵𝑋) → (ran (𝑦𝐽 ↦ (𝑦𝐴)) ↾t 𝐵) = ran (𝑥 ∈ ran (𝑦𝐽 ↦ (𝑦𝐴)) ↦ (𝑥𝐵)))
2118, 19, 20syl2anc 584 . . 3 ((𝐽𝑉𝐴𝑊𝐵𝑋) → (ran (𝑦𝐽 ↦ (𝑦𝐴)) ↾t 𝐵) = ran (𝑥 ∈ ran (𝑦𝐽 ↦ (𝑦𝐴)) ↦ (𝑥𝐵)))
2216, 21eqtrd 2777 . 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 17471 . . 3 ((𝐽𝑉 ∧ (𝐴𝐵) ∈ V) → (𝐽t (𝐴𝐵)) = ran (𝑦𝐽 ↦ (𝑦 ∩ (𝐴𝐵))))
2723, 25, 26syl2anc 584 . 2 ((𝐽𝑉𝐴𝑊𝐵𝑋) → (𝐽t (𝐴𝐵)) = ran (𝑦𝐽 ↦ (𝑦 ∩ (𝐴𝐵))))
2813, 22, 273eqtr4a 2803 1 ((𝐽𝑉𝐴𝑊𝐵𝑋) → ((𝐽t 𝐴) ↾t 𝐵) = (𝐽t (𝐴𝐵)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  w3a 1087   = wceq 1540  wcel 2108  {cab 2714  wrex 3070  Vcvv 3480  cin 3950  cmpt 5225  ran crn 5686  (class class class)co 7431  t crest 17465
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-rest 17467
This theorem is referenced by:  restabs  23173  restin  23174  resstopn  23194  ressuss  24271  smfres  46805
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