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Theorem ressinbas 16130
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 3344 . 2 (𝐴𝑋𝐴 ∈ V)
2 eqid 2752 . . . . . . 7 (𝑊s 𝐴) = (𝑊s 𝐴)
3 ressid.1 . . . . . . 7 𝐵 = (Base‘𝑊)
42, 3ressid2 16122 . . . . . 6 ((𝐵𝐴𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊s 𝐴) = 𝑊)
5 ssid 3757 . . . . . . . 8 𝐵𝐵
6 incom 3940 . . . . . . . . 9 (𝐴𝐵) = (𝐵𝐴)
7 df-ss 3721 . . . . . . . . . 10 (𝐵𝐴 ↔ (𝐵𝐴) = 𝐵)
87biimpi 206 . . . . . . . . 9 (𝐵𝐴 → (𝐵𝐴) = 𝐵)
96, 8syl5eq 2798 . . . . . . . 8 (𝐵𝐴 → (𝐴𝐵) = 𝐵)
105, 9syl5sseqr 3787 . . . . . . 7 (𝐵𝐴𝐵 ⊆ (𝐴𝐵))
11 elex 3344 . . . . . . 7 (𝑊 ∈ V → 𝑊 ∈ V)
12 inex1g 4945 . . . . . . 7 (𝐴 ∈ V → (𝐴𝐵) ∈ V)
13 eqid 2752 . . . . . . . 8 (𝑊s (𝐴𝐵)) = (𝑊s (𝐴𝐵))
1413, 3ressid2 16122 . . . . . . 7 ((𝐵 ⊆ (𝐴𝐵) ∧ 𝑊 ∈ V ∧ (𝐴𝐵) ∈ V) → (𝑊s (𝐴𝐵)) = 𝑊)
1510, 11, 12, 14syl3an 1163 . . . . . 6 ((𝐵𝐴𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊s (𝐴𝐵)) = 𝑊)
164, 15eqtr4d 2789 . . . . 5 ((𝐵𝐴𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊s 𝐴) = (𝑊s (𝐴𝐵)))
17163expb 1113 . . . 4 ((𝐵𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ V)) → (𝑊s 𝐴) = (𝑊s (𝐴𝐵)))
18 inass 3958 . . . . . . . . 9 ((𝐴𝐵) ∩ 𝐵) = (𝐴 ∩ (𝐵𝐵))
19 inidm 3957 . . . . . . . . . 10 (𝐵𝐵) = 𝐵
2019ineq2i 3946 . . . . . . . . 9 (𝐴 ∩ (𝐵𝐵)) = (𝐴𝐵)
2118, 20eqtr2i 2775 . . . . . . . 8 (𝐴𝐵) = ((𝐴𝐵) ∩ 𝐵)
2221opeq2i 4549 . . . . . . 7 ⟨(Base‘ndx), (𝐴𝐵)⟩ = ⟨(Base‘ndx), ((𝐴𝐵) ∩ 𝐵)⟩
2322oveq2i 6816 . . . . . 6 (𝑊 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩) = (𝑊 sSet ⟨(Base‘ndx), ((𝐴𝐵) ∩ 𝐵)⟩)
242, 3ressval2 16123 . . . . . 6 ((¬ 𝐵𝐴𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊s 𝐴) = (𝑊 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩))
25 inss1 3968 . . . . . . . . 9 (𝐴𝐵) ⊆ 𝐴
26 sstr 3744 . . . . . . . . 9 ((𝐵 ⊆ (𝐴𝐵) ∧ (𝐴𝐵) ⊆ 𝐴) → 𝐵𝐴)
2725, 26mpan2 709 . . . . . . . 8 (𝐵 ⊆ (𝐴𝐵) → 𝐵𝐴)
2827con3i 150 . . . . . . 7 𝐵𝐴 → ¬ 𝐵 ⊆ (𝐴𝐵))
2913, 3ressval2 16123 . . . . . . 7 ((¬ 𝐵 ⊆ (𝐴𝐵) ∧ 𝑊 ∈ V ∧ (𝐴𝐵) ∈ V) → (𝑊s (𝐴𝐵)) = (𝑊 sSet ⟨(Base‘ndx), ((𝐴𝐵) ∩ 𝐵)⟩))
3028, 11, 12, 29syl3an 1163 . . . . . 6 ((¬ 𝐵𝐴𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊s (𝐴𝐵)) = (𝑊 sSet ⟨(Base‘ndx), ((𝐴𝐵) ∩ 𝐵)⟩))
3123, 24, 303eqtr4a 2812 . . . . 5 ((¬ 𝐵𝐴𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊s 𝐴) = (𝑊s (𝐴𝐵)))
32313expb 1113 . . . 4 ((¬ 𝐵𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ V)) → (𝑊s 𝐴) = (𝑊s (𝐴𝐵)))
3317, 32pm2.61ian 866 . . 3 ((𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊s 𝐴) = (𝑊s (𝐴𝐵)))
34 reldmress 16120 . . . . . 6 Rel dom ↾s
3534ovprc1 6839 . . . . 5 𝑊 ∈ V → (𝑊s 𝐴) = ∅)
3634ovprc1 6839 . . . . 5 𝑊 ∈ V → (𝑊s (𝐴𝐵)) = ∅)
3735, 36eqtr4d 2789 . . . 4 𝑊 ∈ V → (𝑊s 𝐴) = (𝑊s (𝐴𝐵)))
3837adantr 472 . . 3 ((¬ 𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊s 𝐴) = (𝑊s (𝐴𝐵)))
3933, 38pm2.61ian 866 . 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 1072   = wceq 1624  wcel 2131  Vcvv 3332  cin 3706  wss 3707  c0 4050  cop 4319  cfv 6041  (class class class)co 6805  ndxcnx 16048   sSet csts 16049  Basecbs 16051  s cress 16052
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-iota 6004  df-fun 6043  df-fv 6049  df-ov 6808  df-oprab 6809  df-mpt2 6810  df-ress 16059
This theorem is referenced by:  ressress  16132  rescabs  16686  resscat  16705  funcres2c  16754  ressffth  16791  cphsubrglem  23169  suborng  30116
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