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Theorem ressinbas 17194
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 3491 . 2 (𝐴 ∈ 𝑋 β†’ 𝐴 ∈ V)
2 eqid 2730 . . . . . . 7 (π‘Š β†Ύs 𝐴) = (π‘Š β†Ύs 𝐴)
3 ressid.1 . . . . . . 7 𝐡 = (Baseβ€˜π‘Š)
42, 3ressid2 17181 . . . . . 6 ((𝐡 βŠ† 𝐴 ∧ π‘Š ∈ V ∧ 𝐴 ∈ V) β†’ (π‘Š β†Ύs 𝐴) = π‘Š)
5 ssid 4003 . . . . . . . 8 𝐡 βŠ† 𝐡
6 incom 4200 . . . . . . . . 9 (𝐴 ∩ 𝐡) = (𝐡 ∩ 𝐴)
7 df-ss 3964 . . . . . . . . . 10 (𝐡 βŠ† 𝐴 ↔ (𝐡 ∩ 𝐴) = 𝐡)
87biimpi 215 . . . . . . . . 9 (𝐡 βŠ† 𝐴 β†’ (𝐡 ∩ 𝐴) = 𝐡)
96, 8eqtrid 2782 . . . . . . . 8 (𝐡 βŠ† 𝐴 β†’ (𝐴 ∩ 𝐡) = 𝐡)
105, 9sseqtrrid 4034 . . . . . . 7 (𝐡 βŠ† 𝐴 β†’ 𝐡 βŠ† (𝐴 ∩ 𝐡))
11 elex 3491 . . . . . . 7 (π‘Š ∈ V β†’ π‘Š ∈ V)
12 inex1g 5318 . . . . . . 7 (𝐴 ∈ V β†’ (𝐴 ∩ 𝐡) ∈ V)
13 eqid 2730 . . . . . . . 8 (π‘Š β†Ύs (𝐴 ∩ 𝐡)) = (π‘Š β†Ύs (𝐴 ∩ 𝐡))
1413, 3ressid2 17181 . . . . . . 7 ((𝐡 βŠ† (𝐴 ∩ 𝐡) ∧ π‘Š ∈ V ∧ (𝐴 ∩ 𝐡) ∈ V) β†’ (π‘Š β†Ύs (𝐴 ∩ 𝐡)) = π‘Š)
1510, 11, 12, 14syl3an 1158 . . . . . 6 ((𝐡 βŠ† 𝐴 ∧ π‘Š ∈ V ∧ 𝐴 ∈ V) β†’ (π‘Š β†Ύs (𝐴 ∩ 𝐡)) = π‘Š)
164, 15eqtr4d 2773 . . . . 5 ((𝐡 βŠ† 𝐴 ∧ π‘Š ∈ V ∧ 𝐴 ∈ V) β†’ (π‘Š β†Ύs 𝐴) = (π‘Š β†Ύs (𝐴 ∩ 𝐡)))
17163expb 1118 . . . 4 ((𝐡 βŠ† 𝐴 ∧ (π‘Š ∈ V ∧ 𝐴 ∈ V)) β†’ (π‘Š β†Ύs 𝐴) = (π‘Š β†Ύs (𝐴 ∩ 𝐡)))
18 inass 4218 . . . . . . . . 9 ((𝐴 ∩ 𝐡) ∩ 𝐡) = (𝐴 ∩ (𝐡 ∩ 𝐡))
19 inidm 4217 . . . . . . . . . 10 (𝐡 ∩ 𝐡) = 𝐡
2019ineq2i 4208 . . . . . . . . 9 (𝐴 ∩ (𝐡 ∩ 𝐡)) = (𝐴 ∩ 𝐡)
2118, 20eqtr2i 2759 . . . . . . . 8 (𝐴 ∩ 𝐡) = ((𝐴 ∩ 𝐡) ∩ 𝐡)
2221opeq2i 4876 . . . . . . 7 ⟨(Baseβ€˜ndx), (𝐴 ∩ 𝐡)⟩ = ⟨(Baseβ€˜ndx), ((𝐴 ∩ 𝐡) ∩ 𝐡)⟩
2322oveq2i 7422 . . . . . 6 (π‘Š sSet ⟨(Baseβ€˜ndx), (𝐴 ∩ 𝐡)⟩) = (π‘Š sSet ⟨(Baseβ€˜ndx), ((𝐴 ∩ 𝐡) ∩ 𝐡)⟩)
242, 3ressval2 17182 . . . . . 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 4227 . . . . . . . . 9 (𝐴 ∩ 𝐡) βŠ† 𝐴
26 sstr 3989 . . . . . . . . 9 ((𝐡 βŠ† (𝐴 ∩ 𝐡) ∧ (𝐴 ∩ 𝐡) βŠ† 𝐴) β†’ 𝐡 βŠ† 𝐴)
2725, 26mpan2 687 . . . . . . . 8 (𝐡 βŠ† (𝐴 ∩ 𝐡) β†’ 𝐡 βŠ† 𝐴)
2827con3i 154 . . . . . . 7 (Β¬ 𝐡 βŠ† 𝐴 β†’ Β¬ 𝐡 βŠ† (𝐴 ∩ 𝐡))
2913, 3ressval2 17182 . . . . . . 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 1158 . . . . . 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 2796 . . . . 5 ((Β¬ 𝐡 βŠ† 𝐴 ∧ π‘Š ∈ V ∧ 𝐴 ∈ V) β†’ (π‘Š β†Ύs 𝐴) = (π‘Š β†Ύs (𝐴 ∩ 𝐡)))
32313expb 1118 . . . 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 808 . . 3 ((π‘Š ∈ V ∧ 𝐴 ∈ V) β†’ (π‘Š β†Ύs 𝐴) = (π‘Š β†Ύs (𝐴 ∩ 𝐡)))
34 reldmress 17179 . . . . . 6 Rel dom β†Ύs
3534ovprc1 7450 . . . . 5 (Β¬ π‘Š ∈ V β†’ (π‘Š β†Ύs 𝐴) = βˆ…)
3634ovprc1 7450 . . . . 5 (Β¬ π‘Š ∈ V β†’ (π‘Š β†Ύs (𝐴 ∩ 𝐡)) = βˆ…)
3735, 36eqtr4d 2773 . . . 4 (Β¬ π‘Š ∈ V β†’ (π‘Š β†Ύs 𝐴) = (π‘Š β†Ύs (𝐴 ∩ 𝐡)))
3837adantr 479 . . 3 ((Β¬ π‘Š ∈ V ∧ 𝐴 ∈ V) β†’ (π‘Š β†Ύs 𝐴) = (π‘Š β†Ύs (𝐴 ∩ 𝐡)))
3933, 38pm2.61ian 808 . 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 394   ∧ w3a 1085   = wceq 1539   ∈ wcel 2104  Vcvv 3472   ∩ cin 3946   βŠ† wss 3947  βˆ…c0 4321  βŸ¨cop 4633  β€˜cfv 6542  (class class class)co 7411   sSet csts 17100  ndxcnx 17130  Basecbs 17148   β†Ύs cress 17177
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6494  df-fun 6544  df-fv 6550  df-ov 7414  df-oprab 7415  df-mpo 7416  df-ress 17178
This theorem is referenced by:  ressress  17197  rescabs  17786  rescabsOLD  17787  resscat  17806  funcres2c  17856  ressffth  17893  cphsubrglem  24925  suborng  32703
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