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Theorem ressinbas 17195
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 3492 . 2 (𝐴 ∈ 𝑋 β†’ 𝐴 ∈ V)
2 eqid 2731 . . . . . . 7 (π‘Š β†Ύs 𝐴) = (π‘Š β†Ύs 𝐴)
3 ressid.1 . . . . . . 7 𝐡 = (Baseβ€˜π‘Š)
42, 3ressid2 17182 . . . . . 6 ((𝐡 βŠ† 𝐴 ∧ π‘Š ∈ V ∧ 𝐴 ∈ V) β†’ (π‘Š β†Ύs 𝐴) = π‘Š)
5 ssid 4004 . . . . . . . 8 𝐡 βŠ† 𝐡
6 incom 4201 . . . . . . . . 9 (𝐴 ∩ 𝐡) = (𝐡 ∩ 𝐴)
7 df-ss 3965 . . . . . . . . . 10 (𝐡 βŠ† 𝐴 ↔ (𝐡 ∩ 𝐴) = 𝐡)
87biimpi 215 . . . . . . . . 9 (𝐡 βŠ† 𝐴 β†’ (𝐡 ∩ 𝐴) = 𝐡)
96, 8eqtrid 2783 . . . . . . . 8 (𝐡 βŠ† 𝐴 β†’ (𝐴 ∩ 𝐡) = 𝐡)
105, 9sseqtrrid 4035 . . . . . . 7 (𝐡 βŠ† 𝐴 β†’ 𝐡 βŠ† (𝐴 ∩ 𝐡))
11 elex 3492 . . . . . . 7 (π‘Š ∈ V β†’ π‘Š ∈ V)
12 inex1g 5319 . . . . . . 7 (𝐴 ∈ V β†’ (𝐴 ∩ 𝐡) ∈ V)
13 eqid 2731 . . . . . . . 8 (π‘Š β†Ύs (𝐴 ∩ 𝐡)) = (π‘Š β†Ύs (𝐴 ∩ 𝐡))
1413, 3ressid2 17182 . . . . . . 7 ((𝐡 βŠ† (𝐴 ∩ 𝐡) ∧ π‘Š ∈ V ∧ (𝐴 ∩ 𝐡) ∈ V) β†’ (π‘Š β†Ύs (𝐴 ∩ 𝐡)) = π‘Š)
1510, 11, 12, 14syl3an 1159 . . . . . 6 ((𝐡 βŠ† 𝐴 ∧ π‘Š ∈ V ∧ 𝐴 ∈ V) β†’ (π‘Š β†Ύs (𝐴 ∩ 𝐡)) = π‘Š)
164, 15eqtr4d 2774 . . . . 5 ((𝐡 βŠ† 𝐴 ∧ π‘Š ∈ V ∧ 𝐴 ∈ V) β†’ (π‘Š β†Ύs 𝐴) = (π‘Š β†Ύs (𝐴 ∩ 𝐡)))
17163expb 1119 . . . 4 ((𝐡 βŠ† 𝐴 ∧ (π‘Š ∈ V ∧ 𝐴 ∈ V)) β†’ (π‘Š β†Ύs 𝐴) = (π‘Š β†Ύs (𝐴 ∩ 𝐡)))
18 inass 4219 . . . . . . . . 9 ((𝐴 ∩ 𝐡) ∩ 𝐡) = (𝐴 ∩ (𝐡 ∩ 𝐡))
19 inidm 4218 . . . . . . . . . 10 (𝐡 ∩ 𝐡) = 𝐡
2019ineq2i 4209 . . . . . . . . 9 (𝐴 ∩ (𝐡 ∩ 𝐡)) = (𝐴 ∩ 𝐡)
2118, 20eqtr2i 2760 . . . . . . . 8 (𝐴 ∩ 𝐡) = ((𝐴 ∩ 𝐡) ∩ 𝐡)
2221opeq2i 4877 . . . . . . 7 ⟨(Baseβ€˜ndx), (𝐴 ∩ 𝐡)⟩ = ⟨(Baseβ€˜ndx), ((𝐴 ∩ 𝐡) ∩ 𝐡)⟩
2322oveq2i 7423 . . . . . 6 (π‘Š sSet ⟨(Baseβ€˜ndx), (𝐴 ∩ 𝐡)⟩) = (π‘Š sSet ⟨(Baseβ€˜ndx), ((𝐴 ∩ 𝐡) ∩ 𝐡)⟩)
242, 3ressval2 17183 . . . . . 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 4228 . . . . . . . . 9 (𝐴 ∩ 𝐡) βŠ† 𝐴
26 sstr 3990 . . . . . . . . 9 ((𝐡 βŠ† (𝐴 ∩ 𝐡) ∧ (𝐴 ∩ 𝐡) βŠ† 𝐴) β†’ 𝐡 βŠ† 𝐴)
2725, 26mpan2 688 . . . . . . . 8 (𝐡 βŠ† (𝐴 ∩ 𝐡) β†’ 𝐡 βŠ† 𝐴)
2827con3i 154 . . . . . . 7 (Β¬ 𝐡 βŠ† 𝐴 β†’ Β¬ 𝐡 βŠ† (𝐴 ∩ 𝐡))
2913, 3ressval2 17183 . . . . . . 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 1159 . . . . . 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 2797 . . . . 5 ((Β¬ 𝐡 βŠ† 𝐴 ∧ π‘Š ∈ V ∧ 𝐴 ∈ V) β†’ (π‘Š β†Ύs 𝐴) = (π‘Š β†Ύs (𝐴 ∩ 𝐡)))
32313expb 1119 . . . 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 809 . . 3 ((π‘Š ∈ V ∧ 𝐴 ∈ V) β†’ (π‘Š β†Ύs 𝐴) = (π‘Š β†Ύs (𝐴 ∩ 𝐡)))
34 reldmress 17180 . . . . . 6 Rel dom β†Ύs
3534ovprc1 7451 . . . . 5 (Β¬ π‘Š ∈ V β†’ (π‘Š β†Ύs 𝐴) = βˆ…)
3634ovprc1 7451 . . . . 5 (Β¬ π‘Š ∈ V β†’ (π‘Š β†Ύs (𝐴 ∩ 𝐡)) = βˆ…)
3735, 36eqtr4d 2774 . . . 4 (Β¬ π‘Š ∈ V β†’ (π‘Š β†Ύs 𝐴) = (π‘Š β†Ύs (𝐴 ∩ 𝐡)))
3837adantr 480 . . 3 ((Β¬ π‘Š ∈ V ∧ 𝐴 ∈ V) β†’ (π‘Š β†Ύs 𝐴) = (π‘Š β†Ύs (𝐴 ∩ 𝐡)))
3933, 38pm2.61ian 809 . 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 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  Vcvv 3473   ∩ cin 3947   βŠ† wss 3948  βˆ…c0 4322  βŸ¨cop 4634  β€˜cfv 6543  (class class class)co 7412   sSet csts 17101  ndxcnx 17131  Basecbs 17149   β†Ύs cress 17178
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-iota 6495  df-fun 6545  df-fv 6551  df-ov 7415  df-oprab 7416  df-mpo 7417  df-ress 17179
This theorem is referenced by:  ressress  17198  rescabs  17787  rescabsOLD  17788  resscat  17807  funcres2c  17857  ressffth  17894  cphsubrglem  24926  suborng  32704
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