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Theorem ressinbas 15763
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 3185 . 2 (𝐴𝑋𝐴 ∈ V)
2 eqid 2610 . . . . . . 7 (𝑊s 𝐴) = (𝑊s 𝐴)
3 ressid.1 . . . . . . 7 𝐵 = (Base‘𝑊)
42, 3ressid2 15755 . . . . . 6 ((𝐵𝐴𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊s 𝐴) = 𝑊)
5 ssid 3587 . . . . . . . 8 𝐵𝐵
6 incom 3767 . . . . . . . . 9 (𝐴𝐵) = (𝐵𝐴)
7 df-ss 3554 . . . . . . . . . 10 (𝐵𝐴 ↔ (𝐵𝐴) = 𝐵)
87biimpi 205 . . . . . . . . 9 (𝐵𝐴 → (𝐵𝐴) = 𝐵)
96, 8syl5eq 2656 . . . . . . . 8 (𝐵𝐴 → (𝐴𝐵) = 𝐵)
105, 9syl5sseqr 3617 . . . . . . 7 (𝐵𝐴𝐵 ⊆ (𝐴𝐵))
11 elex 3185 . . . . . . 7 (𝑊 ∈ V → 𝑊 ∈ V)
12 inex1g 4729 . . . . . . 7 (𝐴 ∈ V → (𝐴𝐵) ∈ V)
13 eqid 2610 . . . . . . . 8 (𝑊s (𝐴𝐵)) = (𝑊s (𝐴𝐵))
1413, 3ressid2 15755 . . . . . . 7 ((𝐵 ⊆ (𝐴𝐵) ∧ 𝑊 ∈ V ∧ (𝐴𝐵) ∈ V) → (𝑊s (𝐴𝐵)) = 𝑊)
1510, 11, 12, 14syl3an 1360 . . . . . 6 ((𝐵𝐴𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊s (𝐴𝐵)) = 𝑊)
164, 15eqtr4d 2647 . . . . 5 ((𝐵𝐴𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊s 𝐴) = (𝑊s (𝐴𝐵)))
17163expb 1258 . . . 4 ((𝐵𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ V)) → (𝑊s 𝐴) = (𝑊s (𝐴𝐵)))
18 inass 3785 . . . . . . . . 9 ((𝐴𝐵) ∩ 𝐵) = (𝐴 ∩ (𝐵𝐵))
19 inidm 3784 . . . . . . . . . 10 (𝐵𝐵) = 𝐵
2019ineq2i 3773 . . . . . . . . 9 (𝐴 ∩ (𝐵𝐵)) = (𝐴𝐵)
2118, 20eqtr2i 2633 . . . . . . . 8 (𝐴𝐵) = ((𝐴𝐵) ∩ 𝐵)
2221opeq2i 4344 . . . . . . 7 ⟨(Base‘ndx), (𝐴𝐵)⟩ = ⟨(Base‘ndx), ((𝐴𝐵) ∩ 𝐵)⟩
2322oveq2i 6560 . . . . . 6 (𝑊 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩) = (𝑊 sSet ⟨(Base‘ndx), ((𝐴𝐵) ∩ 𝐵)⟩)
242, 3ressval2 15756 . . . . . 6 ((¬ 𝐵𝐴𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊s 𝐴) = (𝑊 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩))
25 inss1 3795 . . . . . . . . 9 (𝐴𝐵) ⊆ 𝐴
26 sstr 3576 . . . . . . . . 9 ((𝐵 ⊆ (𝐴𝐵) ∧ (𝐴𝐵) ⊆ 𝐴) → 𝐵𝐴)
2725, 26mpan2 703 . . . . . . . 8 (𝐵 ⊆ (𝐴𝐵) → 𝐵𝐴)
2827con3i 149 . . . . . . 7 𝐵𝐴 → ¬ 𝐵 ⊆ (𝐴𝐵))
2913, 3ressval2 15756 . . . . . . 7 ((¬ 𝐵 ⊆ (𝐴𝐵) ∧ 𝑊 ∈ V ∧ (𝐴𝐵) ∈ V) → (𝑊s (𝐴𝐵)) = (𝑊 sSet ⟨(Base‘ndx), ((𝐴𝐵) ∩ 𝐵)⟩))
3028, 11, 12, 29syl3an 1360 . . . . . 6 ((¬ 𝐵𝐴𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊s (𝐴𝐵)) = (𝑊 sSet ⟨(Base‘ndx), ((𝐴𝐵) ∩ 𝐵)⟩))
3123, 24, 303eqtr4a 2670 . . . . 5 ((¬ 𝐵𝐴𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊s 𝐴) = (𝑊s (𝐴𝐵)))
32313expb 1258 . . . 4 ((¬ 𝐵𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ V)) → (𝑊s 𝐴) = (𝑊s (𝐴𝐵)))
3317, 32pm2.61ian 827 . . 3 ((𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊s 𝐴) = (𝑊s (𝐴𝐵)))
34 reldmress 15753 . . . . . 6 Rel dom ↾s
3534ovprc1 6582 . . . . 5 𝑊 ∈ V → (𝑊s 𝐴) = ∅)
3634ovprc1 6582 . . . . 5 𝑊 ∈ V → (𝑊s (𝐴𝐵)) = ∅)
3735, 36eqtr4d 2647 . . . 4 𝑊 ∈ V → (𝑊s 𝐴) = (𝑊s (𝐴𝐵)))
3837adantr 480 . . 3 ((¬ 𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊s 𝐴) = (𝑊s (𝐴𝐵)))
3933, 38pm2.61ian 827 . 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 383  w3a 1031   = wceq 1475  wcel 1977  Vcvv 3173  cin 3539  wss 3540  c0 3874  cop 4131  cfv 5804  (class class class)co 6549  ndxcnx 15692   sSet csts 15693  Basecbs 15695  s cress 15696
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-ress 15702
This theorem is referenced by:  ressress  15765  rescabs  16316  resscat  16335  funcres2c  16384  ressffth  16421  cphsubrglem  22785  suborng  29146
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