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Theorem restco 12359
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 2689 . . . . 5 𝑦 ∈ V
21inex1 4062 . . . 4 (𝑦𝐴) ∈ V
3 ineq1 3270 . . . . 5 (𝑥 = (𝑦𝐴) → (𝑥𝐵) = ((𝑦𝐴) ∩ 𝐵))
4 inass 3286 . . . . 5 ((𝑦𝐴) ∩ 𝐵) = (𝑦 ∩ (𝐴𝐵))
53, 4syl6eq 2188 . . . 4 (𝑥 = (𝑦𝐴) → (𝑥𝐵) = (𝑦 ∩ (𝐴𝐵)))
62, 5abrexco 5660 . . 3 {𝑧 ∣ ∃𝑥 ∈ {𝑤 ∣ ∃𝑦𝐽 𝑤 = (𝑦𝐴)}𝑧 = (𝑥𝐵)} = {𝑧 ∣ ∃𝑦𝐽 𝑧 = (𝑦 ∩ (𝐴𝐵))}
7 eqid 2139 . . . . . 6 (𝑦𝐽 ↦ (𝑦𝐴)) = (𝑦𝐽 ↦ (𝑦𝐴))
87rnmpt 4787 . . . . 5 ran (𝑦𝐽 ↦ (𝑦𝐴)) = {𝑤 ∣ ∃𝑦𝐽 𝑤 = (𝑦𝐴)}
9 mpteq1 4012 . . . . 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 4787 . . 3 ran (𝑥 ∈ ran (𝑦𝐽 ↦ (𝑦𝐴)) ↦ (𝑥𝐵)) = {𝑧 ∣ ∃𝑥 ∈ {𝑤 ∣ ∃𝑦𝐽 𝑤 = (𝑦𝐴)}𝑧 = (𝑥𝐵)}
12 eqid 2139 . . . 4 (𝑦𝐽 ↦ (𝑦 ∩ (𝐴𝐵))) = (𝑦𝐽 ↦ (𝑦 ∩ (𝐴𝐵)))
1312rnmpt 4787 . . 3 ran (𝑦𝐽 ↦ (𝑦 ∩ (𝐴𝐵))) = {𝑧 ∣ ∃𝑦𝐽 𝑧 = (𝑦 ∩ (𝐴𝐵))}
146, 11, 133eqtr4i 2170 . 2 ran (𝑥 ∈ ran (𝑦𝐽 ↦ (𝑦𝐴)) ↦ (𝑥𝐵)) = ran (𝑦𝐽 ↦ (𝑦 ∩ (𝐴𝐵)))
15 restval 12142 . . . . 5 ((𝐽𝑉𝐴𝑊) → (𝐽t 𝐴) = ran (𝑦𝐽 ↦ (𝑦𝐴)))
16153adant3 1001 . . . 4 ((𝐽𝑉𝐴𝑊𝐵𝑋) → (𝐽t 𝐴) = ran (𝑦𝐽 ↦ (𝑦𝐴)))
1716oveq1d 5789 . . 3 ((𝐽𝑉𝐴𝑊𝐵𝑋) → ((𝐽t 𝐴) ↾t 𝐵) = (ran (𝑦𝐽 ↦ (𝑦𝐴)) ↾t 𝐵))
18 restfn 12140 . . . . . 6 t Fn (V × V)
19 simp1 981 . . . . . . 7 ((𝐽𝑉𝐴𝑊𝐵𝑋) → 𝐽𝑉)
2019elexd 2699 . . . . . 6 ((𝐽𝑉𝐴𝑊𝐵𝑋) → 𝐽 ∈ V)
21 simp2 982 . . . . . . 7 ((𝐽𝑉𝐴𝑊𝐵𝑋) → 𝐴𝑊)
2221elexd 2699 . . . . . 6 ((𝐽𝑉𝐴𝑊𝐵𝑋) → 𝐴 ∈ V)
23 fnovex 5804 . . . . . 6 (( ↾t Fn (V × V) ∧ 𝐽 ∈ V ∧ 𝐴 ∈ V) → (𝐽t 𝐴) ∈ V)
2418, 20, 22, 23mp3an2i 1320 . . . . 5 ((𝐽𝑉𝐴𝑊𝐵𝑋) → (𝐽t 𝐴) ∈ V)
2516, 24eqeltrrd 2217 . . . 4 ((𝐽𝑉𝐴𝑊𝐵𝑋) → ran (𝑦𝐽 ↦ (𝑦𝐴)) ∈ V)
26 simp3 983 . . . 4 ((𝐽𝑉𝐴𝑊𝐵𝑋) → 𝐵𝑋)
27 restval 12142 . . . 4 ((ran (𝑦𝐽 ↦ (𝑦𝐴)) ∈ V ∧ 𝐵𝑋) → (ran (𝑦𝐽 ↦ (𝑦𝐴)) ↾t 𝐵) = ran (𝑥 ∈ ran (𝑦𝐽 ↦ (𝑦𝐴)) ↦ (𝑥𝐵)))
2825, 26, 27syl2anc 408 . . 3 ((𝐽𝑉𝐴𝑊𝐵𝑋) → (ran (𝑦𝐽 ↦ (𝑦𝐴)) ↾t 𝐵) = ran (𝑥 ∈ ran (𝑦𝐽 ↦ (𝑦𝐴)) ↦ (𝑥𝐵)))
2917, 28eqtrd 2172 . 2 ((𝐽𝑉𝐴𝑊𝐵𝑋) → ((𝐽t 𝐴) ↾t 𝐵) = ran (𝑥 ∈ ran (𝑦𝐽 ↦ (𝑦𝐴)) ↦ (𝑥𝐵)))
30 inex1g 4064 . . . 4 (𝐴𝑊 → (𝐴𝐵) ∈ V)
31303ad2ant2 1003 . . 3 ((𝐽𝑉𝐴𝑊𝐵𝑋) → (𝐴𝐵) ∈ V)
32 restval 12142 . . 3 ((𝐽𝑉 ∧ (𝐴𝐵) ∈ V) → (𝐽t (𝐴𝐵)) = ran (𝑦𝐽 ↦ (𝑦 ∩ (𝐴𝐵))))
3319, 31, 32syl2anc 408 . 2 ((𝐽𝑉𝐴𝑊𝐵𝑋) → (𝐽t (𝐴𝐵)) = ran (𝑦𝐽 ↦ (𝑦 ∩ (𝐴𝐵))))
3414, 29, 333eqtr4a 2198 1 ((𝐽𝑉𝐴𝑊𝐵𝑋) → ((𝐽t 𝐴) ↾t 𝐵) = (𝐽t (𝐴𝐵)))
Colors of variables: wff set class
Syntax hints:  wi 4  w3a 962   = wceq 1331  wcel 1480  {cab 2125  wrex 2417  Vcvv 2686  cin 3070  cmpt 3989   × cxp 4537  ran crn 4540   Fn wfn 5118  (class class class)co 5774  t crest 12136
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-1st 6038  df-2nd 6039  df-rest 12138
This theorem is referenced by:  restabs  12360  restin  12361
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