MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ressinbas Structured version   Visualization version   GIF version

Theorem ressinbas 17172
Description: Restriction only cares about the part of the second set which intersects the base of the first. (Contributed by Stefan O'Rear, 29-Nov-2014.)
Hypothesis
Ref Expression
ressid.1 𝐵 = (Base‘𝑊)
Assertion
Ref Expression
ressinbas (𝐴𝑋 → (𝑊s 𝐴) = (𝑊s (𝐴𝐵)))

Proof of Theorem ressinbas
StepHypRef Expression
1 elex 3491 . 2 (𝐴𝑋𝐴 ∈ V)
2 eqid 2731 . . . . . . 7 (𝑊s 𝐴) = (𝑊s 𝐴)
3 ressid.1 . . . . . . 7 𝐵 = (Base‘𝑊)
42, 3ressid2 17159 . . . . . 6 ((𝐵𝐴𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊s 𝐴) = 𝑊)
5 ssid 4000 . . . . . . . 8 𝐵𝐵
6 incom 4197 . . . . . . . . 9 (𝐴𝐵) = (𝐵𝐴)
7 df-ss 3961 . . . . . . . . . 10 (𝐵𝐴 ↔ (𝐵𝐴) = 𝐵)
87biimpi 215 . . . . . . . . 9 (𝐵𝐴 → (𝐵𝐴) = 𝐵)
96, 8eqtrid 2783 . . . . . . . 8 (𝐵𝐴 → (𝐴𝐵) = 𝐵)
105, 9sseqtrrid 4031 . . . . . . 7 (𝐵𝐴𝐵 ⊆ (𝐴𝐵))
11 elex 3491 . . . . . . 7 (𝑊 ∈ V → 𝑊 ∈ V)
12 inex1g 5312 . . . . . . 7 (𝐴 ∈ V → (𝐴𝐵) ∈ V)
13 eqid 2731 . . . . . . . 8 (𝑊s (𝐴𝐵)) = (𝑊s (𝐴𝐵))
1413, 3ressid2 17159 . . . . . . 7 ((𝐵 ⊆ (𝐴𝐵) ∧ 𝑊 ∈ V ∧ (𝐴𝐵) ∈ V) → (𝑊s (𝐴𝐵)) = 𝑊)
1510, 11, 12, 14syl3an 1160 . . . . . 6 ((𝐵𝐴𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊s (𝐴𝐵)) = 𝑊)
164, 15eqtr4d 2774 . . . . 5 ((𝐵𝐴𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊s 𝐴) = (𝑊s (𝐴𝐵)))
17163expb 1120 . . . 4 ((𝐵𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ V)) → (𝑊s 𝐴) = (𝑊s (𝐴𝐵)))
18 inass 4215 . . . . . . . . 9 ((𝐴𝐵) ∩ 𝐵) = (𝐴 ∩ (𝐵𝐵))
19 inidm 4214 . . . . . . . . . 10 (𝐵𝐵) = 𝐵
2019ineq2i 4205 . . . . . . . . 9 (𝐴 ∩ (𝐵𝐵)) = (𝐴𝐵)
2118, 20eqtr2i 2760 . . . . . . . 8 (𝐴𝐵) = ((𝐴𝐵) ∩ 𝐵)
2221opeq2i 4870 . . . . . . 7 ⟨(Base‘ndx), (𝐴𝐵)⟩ = ⟨(Base‘ndx), ((𝐴𝐵) ∩ 𝐵)⟩
2322oveq2i 7404 . . . . . 6 (𝑊 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩) = (𝑊 sSet ⟨(Base‘ndx), ((𝐴𝐵) ∩ 𝐵)⟩)
242, 3ressval2 17160 . . . . . 6 ((¬ 𝐵𝐴𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊s 𝐴) = (𝑊 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩))
25 inss1 4224 . . . . . . . . 9 (𝐴𝐵) ⊆ 𝐴
26 sstr 3986 . . . . . . . . 9 ((𝐵 ⊆ (𝐴𝐵) ∧ (𝐴𝐵) ⊆ 𝐴) → 𝐵𝐴)
2725, 26mpan2 689 . . . . . . . 8 (𝐵 ⊆ (𝐴𝐵) → 𝐵𝐴)
2827con3i 154 . . . . . . 7 𝐵𝐴 → ¬ 𝐵 ⊆ (𝐴𝐵))
2913, 3ressval2 17160 . . . . . . 7 ((¬ 𝐵 ⊆ (𝐴𝐵) ∧ 𝑊 ∈ V ∧ (𝐴𝐵) ∈ V) → (𝑊s (𝐴𝐵)) = (𝑊 sSet ⟨(Base‘ndx), ((𝐴𝐵) ∩ 𝐵)⟩))
3028, 11, 12, 29syl3an 1160 . . . . . 6 ((¬ 𝐵𝐴𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊s (𝐴𝐵)) = (𝑊 sSet ⟨(Base‘ndx), ((𝐴𝐵) ∩ 𝐵)⟩))
3123, 24, 303eqtr4a 2797 . . . . 5 ((¬ 𝐵𝐴𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊s 𝐴) = (𝑊s (𝐴𝐵)))
32313expb 1120 . . . 4 ((¬ 𝐵𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ V)) → (𝑊s 𝐴) = (𝑊s (𝐴𝐵)))
3317, 32pm2.61ian 810 . . 3 ((𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊s 𝐴) = (𝑊s (𝐴𝐵)))
34 reldmress 17157 . . . . . 6 Rel dom ↾s
3534ovprc1 7432 . . . . 5 𝑊 ∈ V → (𝑊s 𝐴) = ∅)
3634ovprc1 7432 . . . . 5 𝑊 ∈ V → (𝑊s (𝐴𝐵)) = ∅)
3735, 36eqtr4d 2774 . . . 4 𝑊 ∈ V → (𝑊s 𝐴) = (𝑊s (𝐴𝐵)))
3837adantr 481 . . 3 ((¬ 𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊s 𝐴) = (𝑊s (𝐴𝐵)))
3933, 38pm2.61ian 810 . 2 (𝐴 ∈ V → (𝑊s 𝐴) = (𝑊s (𝐴𝐵)))
401, 39syl 17 1 (𝐴𝑋 → (𝑊s 𝐴) = (𝑊s (𝐴𝐵)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  Vcvv 3473  cin 3943  wss 3944  c0 4318  cop 4628  cfv 6532  (class class class)co 7393   sSet csts 17078  ndxcnx 17108  Basecbs 17126  s cress 17155
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3774  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-iota 6484  df-fun 6534  df-fv 6540  df-ov 7396  df-oprab 7397  df-mpo 7398  df-ress 17156
This theorem is referenced by:  ressress  17175  rescabs  17764  rescabsOLD  17765  resscat  17784  funcres2c  17834  ressffth  17871  cphsubrglem  24623  suborng  32295
  Copyright terms: Public domain W3C validator