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Theorem ressinbas 17291
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 3501 . 2 (𝐴𝑋𝐴 ∈ V)
2 eqid 2737 . . . . . . 7 (𝑊s 𝐴) = (𝑊s 𝐴)
3 ressid.1 . . . . . . 7 𝐵 = (Base‘𝑊)
42, 3ressid2 17278 . . . . . 6 ((𝐵𝐴𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊s 𝐴) = 𝑊)
5 ssid 4006 . . . . . . . 8 𝐵𝐵
6 incom 4209 . . . . . . . . 9 (𝐴𝐵) = (𝐵𝐴)
7 dfss2 3969 . . . . . . . . . 10 (𝐵𝐴 ↔ (𝐵𝐴) = 𝐵)
87biimpi 216 . . . . . . . . 9 (𝐵𝐴 → (𝐵𝐴) = 𝐵)
96, 8eqtrid 2789 . . . . . . . 8 (𝐵𝐴 → (𝐴𝐵) = 𝐵)
105, 9sseqtrrid 4027 . . . . . . 7 (𝐵𝐴𝐵 ⊆ (𝐴𝐵))
11 elex 3501 . . . . . . 7 (𝑊 ∈ V → 𝑊 ∈ V)
12 inex1g 5319 . . . . . . 7 (𝐴 ∈ V → (𝐴𝐵) ∈ V)
13 eqid 2737 . . . . . . . 8 (𝑊s (𝐴𝐵)) = (𝑊s (𝐴𝐵))
1413, 3ressid2 17278 . . . . . . 7 ((𝐵 ⊆ (𝐴𝐵) ∧ 𝑊 ∈ V ∧ (𝐴𝐵) ∈ V) → (𝑊s (𝐴𝐵)) = 𝑊)
1510, 11, 12, 14syl3an 1161 . . . . . 6 ((𝐵𝐴𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊s (𝐴𝐵)) = 𝑊)
164, 15eqtr4d 2780 . . . . 5 ((𝐵𝐴𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊s 𝐴) = (𝑊s (𝐴𝐵)))
17163expb 1121 . . . 4 ((𝐵𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ V)) → (𝑊s 𝐴) = (𝑊s (𝐴𝐵)))
18 inass 4228 . . . . . . . . 9 ((𝐴𝐵) ∩ 𝐵) = (𝐴 ∩ (𝐵𝐵))
19 inidm 4227 . . . . . . . . . 10 (𝐵𝐵) = 𝐵
2019ineq2i 4217 . . . . . . . . 9 (𝐴 ∩ (𝐵𝐵)) = (𝐴𝐵)
2118, 20eqtr2i 2766 . . . . . . . 8 (𝐴𝐵) = ((𝐴𝐵) ∩ 𝐵)
2221opeq2i 4877 . . . . . . 7 ⟨(Base‘ndx), (𝐴𝐵)⟩ = ⟨(Base‘ndx), ((𝐴𝐵) ∩ 𝐵)⟩
2322oveq2i 7442 . . . . . 6 (𝑊 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩) = (𝑊 sSet ⟨(Base‘ndx), ((𝐴𝐵) ∩ 𝐵)⟩)
242, 3ressval2 17279 . . . . . 6 ((¬ 𝐵𝐴𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊s 𝐴) = (𝑊 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩))
25 inss1 4237 . . . . . . . . 9 (𝐴𝐵) ⊆ 𝐴
26 sstr 3992 . . . . . . . . 9 ((𝐵 ⊆ (𝐴𝐵) ∧ (𝐴𝐵) ⊆ 𝐴) → 𝐵𝐴)
2725, 26mpan2 691 . . . . . . . 8 (𝐵 ⊆ (𝐴𝐵) → 𝐵𝐴)
2827con3i 154 . . . . . . 7 𝐵𝐴 → ¬ 𝐵 ⊆ (𝐴𝐵))
2913, 3ressval2 17279 . . . . . . 7 ((¬ 𝐵 ⊆ (𝐴𝐵) ∧ 𝑊 ∈ V ∧ (𝐴𝐵) ∈ V) → (𝑊s (𝐴𝐵)) = (𝑊 sSet ⟨(Base‘ndx), ((𝐴𝐵) ∩ 𝐵)⟩))
3028, 11, 12, 29syl3an 1161 . . . . . 6 ((¬ 𝐵𝐴𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊s (𝐴𝐵)) = (𝑊 sSet ⟨(Base‘ndx), ((𝐴𝐵) ∩ 𝐵)⟩))
3123, 24, 303eqtr4a 2803 . . . . 5 ((¬ 𝐵𝐴𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊s 𝐴) = (𝑊s (𝐴𝐵)))
32313expb 1121 . . . 4 ((¬ 𝐵𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ V)) → (𝑊s 𝐴) = (𝑊s (𝐴𝐵)))
3317, 32pm2.61ian 812 . . 3 ((𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊s 𝐴) = (𝑊s (𝐴𝐵)))
34 reldmress 17276 . . . . . 6 Rel dom ↾s
3534ovprc1 7470 . . . . 5 𝑊 ∈ V → (𝑊s 𝐴) = ∅)
3634ovprc1 7470 . . . . 5 𝑊 ∈ V → (𝑊s (𝐴𝐵)) = ∅)
3735, 36eqtr4d 2780 . . . 4 𝑊 ∈ V → (𝑊s 𝐴) = (𝑊s (𝐴𝐵)))
3837adantr 480 . . 3 ((¬ 𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊s 𝐴) = (𝑊s (𝐴𝐵)))
3933, 38pm2.61ian 812 . 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 395  w3a 1087   = wceq 1540  wcel 2108  Vcvv 3480  cin 3950  wss 3951  c0 4333  cop 4632  cfv 6561  (class class class)co 7431   sSet csts 17200  ndxcnx 17230  Basecbs 17247  s cress 17274
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-ress 17275
This theorem is referenced by:  ressress  17293  rescabs  17877  rescabsOLD  17878  resscat  17897  funcres2c  17948  ressffth  17985  cphsubrglem  25211  suborng  33345
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