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Theorem ressinbas 17131
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 𝐡 = (Baseβ€˜π‘Š)
Assertion
Ref Expression
ressinbas (𝐴 ∈ 𝑋 β†’ (π‘Š β†Ύs 𝐴) = (π‘Š β†Ύs (𝐴 ∩ 𝐡)))

Proof of Theorem ressinbas
StepHypRef Expression
1 elex 3462 . 2 (𝐴 ∈ 𝑋 β†’ 𝐴 ∈ V)
2 eqid 2733 . . . . . . 7 (π‘Š β†Ύs 𝐴) = (π‘Š β†Ύs 𝐴)
3 ressid.1 . . . . . . 7 𝐡 = (Baseβ€˜π‘Š)
42, 3ressid2 17121 . . . . . 6 ((𝐡 βŠ† 𝐴 ∧ π‘Š ∈ V ∧ 𝐴 ∈ V) β†’ (π‘Š β†Ύs 𝐴) = π‘Š)
5 ssid 3967 . . . . . . . 8 𝐡 βŠ† 𝐡
6 incom 4162 . . . . . . . . 9 (𝐴 ∩ 𝐡) = (𝐡 ∩ 𝐴)
7 df-ss 3928 . . . . . . . . . 10 (𝐡 βŠ† 𝐴 ↔ (𝐡 ∩ 𝐴) = 𝐡)
87biimpi 215 . . . . . . . . 9 (𝐡 βŠ† 𝐴 β†’ (𝐡 ∩ 𝐴) = 𝐡)
96, 8eqtrid 2785 . . . . . . . 8 (𝐡 βŠ† 𝐴 β†’ (𝐴 ∩ 𝐡) = 𝐡)
105, 9sseqtrrid 3998 . . . . . . 7 (𝐡 βŠ† 𝐴 β†’ 𝐡 βŠ† (𝐴 ∩ 𝐡))
11 elex 3462 . . . . . . 7 (π‘Š ∈ V β†’ π‘Š ∈ V)
12 inex1g 5277 . . . . . . 7 (𝐴 ∈ V β†’ (𝐴 ∩ 𝐡) ∈ V)
13 eqid 2733 . . . . . . . 8 (π‘Š β†Ύs (𝐴 ∩ 𝐡)) = (π‘Š β†Ύs (𝐴 ∩ 𝐡))
1413, 3ressid2 17121 . . . . . . 7 ((𝐡 βŠ† (𝐴 ∩ 𝐡) ∧ π‘Š ∈ V ∧ (𝐴 ∩ 𝐡) ∈ V) β†’ (π‘Š β†Ύs (𝐴 ∩ 𝐡)) = π‘Š)
1510, 11, 12, 14syl3an 1161 . . . . . 6 ((𝐡 βŠ† 𝐴 ∧ π‘Š ∈ V ∧ 𝐴 ∈ V) β†’ (π‘Š β†Ύs (𝐴 ∩ 𝐡)) = π‘Š)
164, 15eqtr4d 2776 . . . . 5 ((𝐡 βŠ† 𝐴 ∧ π‘Š ∈ V ∧ 𝐴 ∈ V) β†’ (π‘Š β†Ύs 𝐴) = (π‘Š β†Ύs (𝐴 ∩ 𝐡)))
17163expb 1121 . . . 4 ((𝐡 βŠ† 𝐴 ∧ (π‘Š ∈ V ∧ 𝐴 ∈ V)) β†’ (π‘Š β†Ύs 𝐴) = (π‘Š β†Ύs (𝐴 ∩ 𝐡)))
18 inass 4180 . . . . . . . . 9 ((𝐴 ∩ 𝐡) ∩ 𝐡) = (𝐴 ∩ (𝐡 ∩ 𝐡))
19 inidm 4179 . . . . . . . . . 10 (𝐡 ∩ 𝐡) = 𝐡
2019ineq2i 4170 . . . . . . . . 9 (𝐴 ∩ (𝐡 ∩ 𝐡)) = (𝐴 ∩ 𝐡)
2118, 20eqtr2i 2762 . . . . . . . 8 (𝐴 ∩ 𝐡) = ((𝐴 ∩ 𝐡) ∩ 𝐡)
2221opeq2i 4835 . . . . . . 7 ⟨(Baseβ€˜ndx), (𝐴 ∩ 𝐡)⟩ = ⟨(Baseβ€˜ndx), ((𝐴 ∩ 𝐡) ∩ 𝐡)⟩
2322oveq2i 7369 . . . . . 6 (π‘Š sSet ⟨(Baseβ€˜ndx), (𝐴 ∩ 𝐡)⟩) = (π‘Š sSet ⟨(Baseβ€˜ndx), ((𝐴 ∩ 𝐡) ∩ 𝐡)⟩)
242, 3ressval2 17122 . . . . . 6 ((Β¬ 𝐡 βŠ† 𝐴 ∧ π‘Š ∈ V ∧ 𝐴 ∈ V) β†’ (π‘Š β†Ύs 𝐴) = (π‘Š sSet ⟨(Baseβ€˜ndx), (𝐴 ∩ 𝐡)⟩))
25 inss1 4189 . . . . . . . . 9 (𝐴 ∩ 𝐡) βŠ† 𝐴
26 sstr 3953 . . . . . . . . 9 ((𝐡 βŠ† (𝐴 ∩ 𝐡) ∧ (𝐴 ∩ 𝐡) βŠ† 𝐴) β†’ 𝐡 βŠ† 𝐴)
2725, 26mpan2 690 . . . . . . . 8 (𝐡 βŠ† (𝐴 ∩ 𝐡) β†’ 𝐡 βŠ† 𝐴)
2827con3i 154 . . . . . . 7 (Β¬ 𝐡 βŠ† 𝐴 β†’ Β¬ 𝐡 βŠ† (𝐴 ∩ 𝐡))
2913, 3ressval2 17122 . . . . . . 7 ((Β¬ 𝐡 βŠ† (𝐴 ∩ 𝐡) ∧ π‘Š ∈ V ∧ (𝐴 ∩ 𝐡) ∈ V) β†’ (π‘Š β†Ύs (𝐴 ∩ 𝐡)) = (π‘Š sSet ⟨(Baseβ€˜ndx), ((𝐴 ∩ 𝐡) ∩ 𝐡)⟩))
3028, 11, 12, 29syl3an 1161 . . . . . 6 ((Β¬ 𝐡 βŠ† 𝐴 ∧ π‘Š ∈ V ∧ 𝐴 ∈ V) β†’ (π‘Š β†Ύs (𝐴 ∩ 𝐡)) = (π‘Š sSet ⟨(Baseβ€˜ndx), ((𝐴 ∩ 𝐡) ∩ 𝐡)⟩))
3123, 24, 303eqtr4a 2799 . . . . 5 ((Β¬ 𝐡 βŠ† 𝐴 ∧ π‘Š ∈ V ∧ 𝐴 ∈ V) β†’ (π‘Š β†Ύs 𝐴) = (π‘Š β†Ύs (𝐴 ∩ 𝐡)))
32313expb 1121 . . . 4 ((Β¬ 𝐡 βŠ† 𝐴 ∧ (π‘Š ∈ V ∧ 𝐴 ∈ V)) β†’ (π‘Š β†Ύs 𝐴) = (π‘Š β†Ύs (𝐴 ∩ 𝐡)))
3317, 32pm2.61ian 811 . . 3 ((π‘Š ∈ V ∧ 𝐴 ∈ V) β†’ (π‘Š β†Ύs 𝐴) = (π‘Š β†Ύs (𝐴 ∩ 𝐡)))
34 reldmress 17119 . . . . . 6 Rel dom β†Ύs
3534ovprc1 7397 . . . . 5 (Β¬ π‘Š ∈ V β†’ (π‘Š β†Ύs 𝐴) = βˆ…)
3634ovprc1 7397 . . . . 5 (Β¬ π‘Š ∈ V β†’ (π‘Š β†Ύs (𝐴 ∩ 𝐡)) = βˆ…)
3735, 36eqtr4d 2776 . . . 4 (Β¬ π‘Š ∈ V β†’ (π‘Š β†Ύs 𝐴) = (π‘Š β†Ύs (𝐴 ∩ 𝐡)))
3837adantr 482 . . 3 ((Β¬ π‘Š ∈ V ∧ 𝐴 ∈ V) β†’ (π‘Š β†Ύs 𝐴) = (π‘Š β†Ύs (𝐴 ∩ 𝐡)))
3933, 38pm2.61ian 811 . 2 (𝐴 ∈ V β†’ (π‘Š β†Ύs 𝐴) = (π‘Š β†Ύs (𝐴 ∩ 𝐡)))
401, 39syl 17 1 (𝐴 ∈ 𝑋 β†’ (π‘Š β†Ύs 𝐴) = (π‘Š β†Ύs (𝐴 ∩ 𝐡)))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  Vcvv 3444   ∩ cin 3910   βŠ† wss 3911  βˆ…c0 4283  βŸ¨cop 4593  β€˜cfv 6497  (class class class)co 7358   sSet csts 17040  ndxcnx 17070  Basecbs 17088   β†Ύs cress 17117
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-iota 6449  df-fun 6499  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-ress 17118
This theorem is referenced by:  ressress  17134  rescabs  17723  rescabsOLD  17724  resscat  17743  funcres2c  17793  ressffth  17830  cphsubrglem  24557  suborng  32157
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