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Theorem ressinbas 16665
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 3416 . 2 (𝐴𝑋𝐴 ∈ V)
2 eqid 2738 . . . . . . 7 (𝑊s 𝐴) = (𝑊s 𝐴)
3 ressid.1 . . . . . . 7 𝐵 = (Base‘𝑊)
42, 3ressid2 16657 . . . . . 6 ((𝐵𝐴𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊s 𝐴) = 𝑊)
5 ssid 3899 . . . . . . . 8 𝐵𝐵
6 incom 4091 . . . . . . . . 9 (𝐴𝐵) = (𝐵𝐴)
7 df-ss 3860 . . . . . . . . . 10 (𝐵𝐴 ↔ (𝐵𝐴) = 𝐵)
87biimpi 219 . . . . . . . . 9 (𝐵𝐴 → (𝐵𝐴) = 𝐵)
96, 8syl5eq 2785 . . . . . . . 8 (𝐵𝐴 → (𝐴𝐵) = 𝐵)
105, 9sseqtrrid 3930 . . . . . . 7 (𝐵𝐴𝐵 ⊆ (𝐴𝐵))
11 elex 3416 . . . . . . 7 (𝑊 ∈ V → 𝑊 ∈ V)
12 inex1g 5187 . . . . . . 7 (𝐴 ∈ V → (𝐴𝐵) ∈ V)
13 eqid 2738 . . . . . . . 8 (𝑊s (𝐴𝐵)) = (𝑊s (𝐴𝐵))
1413, 3ressid2 16657 . . . . . . 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 2776 . . . . 5 ((𝐵𝐴𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊s 𝐴) = (𝑊s (𝐴𝐵)))
17163expb 1121 . . . 4 ((𝐵𝐴 ∧ (𝑊 ∈ V ∧ 𝐴 ∈ V)) → (𝑊s 𝐴) = (𝑊s (𝐴𝐵)))
18 inass 4110 . . . . . . . . 9 ((𝐴𝐵) ∩ 𝐵) = (𝐴 ∩ (𝐵𝐵))
19 inidm 4109 . . . . . . . . . 10 (𝐵𝐵) = 𝐵
2019ineq2i 4100 . . . . . . . . 9 (𝐴 ∩ (𝐵𝐵)) = (𝐴𝐵)
2118, 20eqtr2i 2762 . . . . . . . 8 (𝐴𝐵) = ((𝐴𝐵) ∩ 𝐵)
2221opeq2i 4765 . . . . . . 7 ⟨(Base‘ndx), (𝐴𝐵)⟩ = ⟨(Base‘ndx), ((𝐴𝐵) ∩ 𝐵)⟩
2322oveq2i 7183 . . . . . 6 (𝑊 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩) = (𝑊 sSet ⟨(Base‘ndx), ((𝐴𝐵) ∩ 𝐵)⟩)
242, 3ressval2 16658 . . . . . 6 ((¬ 𝐵𝐴𝑊 ∈ V ∧ 𝐴 ∈ V) → (𝑊s 𝐴) = (𝑊 sSet ⟨(Base‘ndx), (𝐴𝐵)⟩))
25 inss1 4119 . . . . . . . . 9 (𝐴𝐵) ⊆ 𝐴
26 sstr 3885 . . . . . . . . 9 ((𝐵 ⊆ (𝐴𝐵) ∧ (𝐴𝐵) ⊆ 𝐴) → 𝐵𝐴)
2725, 26mpan2 691 . . . . . . . 8 (𝐵 ⊆ (𝐴𝐵) → 𝐵𝐴)
2827con3i 157 . . . . . . 7 𝐵𝐴 → ¬ 𝐵 ⊆ (𝐴𝐵))
2913, 3ressval2 16658 . . . . . . 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 2799 . . . . 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 16655 . . . . . 6 Rel dom ↾s
3534ovprc1 7211 . . . . 5 𝑊 ∈ V → (𝑊s 𝐴) = ∅)
3634ovprc1 7211 . . . . 5 𝑊 ∈ V → (𝑊s (𝐴𝐵)) = ∅)
3735, 36eqtr4d 2776 . . . 4 𝑊 ∈ V → (𝑊s 𝐴) = (𝑊s (𝐴𝐵)))
3837adantr 484 . . 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 399  w3a 1088   = wceq 1542  wcel 2114  Vcvv 3398  cin 3842  wss 3843  c0 4211  cop 4522  cfv 6339  (class class class)co 7172  ndxcnx 16585   sSet csts 16586  Basecbs 16588  s cress 16589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pr 5296
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2540  df-eu 2570  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-ne 2935  df-ral 3058  df-rex 3059  df-rab 3062  df-v 3400  df-sbc 3681  df-dif 3846  df-un 3848  df-in 3850  df-ss 3860  df-nul 4212  df-if 4415  df-sn 4517  df-pr 4519  df-op 4523  df-uni 4797  df-br 5031  df-opab 5093  df-id 5429  df-xp 5531  df-rel 5532  df-cnv 5533  df-co 5534  df-dm 5535  df-iota 6297  df-fun 6341  df-fv 6347  df-ov 7175  df-oprab 7176  df-mpo 7177  df-ress 16596
This theorem is referenced by:  ressress  16667  rescabs  17210  resscat  17229  funcres2c  17278  ressffth  17315  cphsubrglem  23931  suborng  31093
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