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

Theorem resseqnbas 17221
Description: The components of an extensible structure except the base set remain unchanged on a structure restriction. (Contributed by Mario Carneiro, 26-Nov-2014.) (Revised by Mario Carneiro, 2-Dec-2014.) (Revised by AV, 19-Oct-2024.)
Hypotheses
Ref Expression
resseqnbas.r 𝑅 = (𝑊s 𝐴)
resseqnbas.e 𝐶 = (𝐸𝑊)
resseqnbas.f 𝐸 = Slot (𝐸‘ndx)
resseqnbas.n (𝐸‘ndx) ≠ (Base‘ndx)
Assertion
Ref Expression
resseqnbas (𝐴𝑉𝐶 = (𝐸𝑅))

Proof of Theorem resseqnbas
StepHypRef Expression
1 resseqnbas.e . 2 𝐶 = (𝐸𝑊)
2 resseqnbas.r . . . . . . 7 𝑅 = (𝑊s 𝐴)
3 eqid 2728 . . . . . . 7 (Base‘𝑊) = (Base‘𝑊)
42, 3ressid2 17212 . . . . . 6 (((Base‘𝑊) ⊆ 𝐴𝑊 ∈ V ∧ 𝐴𝑉) → 𝑅 = 𝑊)
54fveq2d 6901 . . . . 5 (((Base‘𝑊) ⊆ 𝐴𝑊 ∈ V ∧ 𝐴𝑉) → (𝐸𝑅) = (𝐸𝑊))
653expib 1120 . . . 4 ((Base‘𝑊) ⊆ 𝐴 → ((𝑊 ∈ V ∧ 𝐴𝑉) → (𝐸𝑅) = (𝐸𝑊)))
72, 3ressval2 17213 . . . . . . 7 ((¬ (Base‘𝑊) ⊆ 𝐴𝑊 ∈ V ∧ 𝐴𝑉) → 𝑅 = (𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩))
87fveq2d 6901 . . . . . 6 ((¬ (Base‘𝑊) ⊆ 𝐴𝑊 ∈ V ∧ 𝐴𝑉) → (𝐸𝑅) = (𝐸‘(𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩)))
9 resseqnbas.f . . . . . . 7 𝐸 = Slot (𝐸‘ndx)
10 resseqnbas.n . . . . . . 7 (𝐸‘ndx) ≠ (Base‘ndx)
119, 10setsnid 17177 . . . . . 6 (𝐸𝑊) = (𝐸‘(𝑊 sSet ⟨(Base‘ndx), (𝐴 ∩ (Base‘𝑊))⟩))
128, 11eqtr4di 2786 . . . . 5 ((¬ (Base‘𝑊) ⊆ 𝐴𝑊 ∈ V ∧ 𝐴𝑉) → (𝐸𝑅) = (𝐸𝑊))
13123expib 1120 . . . 4 (¬ (Base‘𝑊) ⊆ 𝐴 → ((𝑊 ∈ V ∧ 𝐴𝑉) → (𝐸𝑅) = (𝐸𝑊)))
146, 13pm2.61i 182 . . 3 ((𝑊 ∈ V ∧ 𝐴𝑉) → (𝐸𝑅) = (𝐸𝑊))
159str0 17157 . . . . . . 7 ∅ = (𝐸‘∅)
1615eqcomi 2737 . . . . . 6 (𝐸‘∅) = ∅
17 reldmress 17210 . . . . . 6 Rel dom ↾s
1816, 2, 17oveqprc 17160 . . . . 5 𝑊 ∈ V → (𝐸𝑊) = (𝐸𝑅))
1918eqcomd 2734 . . . 4 𝑊 ∈ V → (𝐸𝑅) = (𝐸𝑊))
2019adantr 480 . . 3 ((¬ 𝑊 ∈ V ∧ 𝐴𝑉) → (𝐸𝑅) = (𝐸𝑊))
2114, 20pm2.61ian 811 . 2 (𝐴𝑉 → (𝐸𝑅) = (𝐸𝑊))
221, 21eqtr4id 2787 1 (𝐴𝑉𝐶 = (𝐸𝑅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  w3a 1085   = wceq 1534  wcel 2099  wne 2937  Vcvv 3471  cin 3946  wss 3947  c0 4323  cop 4635  cfv 6548  (class class class)co 7420   sSet csts 17131  Slot cslot 17149  ndxcnx 17161  Basecbs 17179  s cress 17208
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-res 5690  df-iota 6500  df-fun 6550  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-sets 17132  df-slot 17150  df-ress 17209
This theorem is referenced by:  ressplusg  17270  ressmulr  17287  ressstarv  17288  resssca  17323  ressvsca  17324  ressip  17325  resstset  17345  ressle  17360  ressunif  17382  ressds  17390  resshom  17399  ressco  17400
  Copyright terms: Public domain W3C validator