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Theorem resvlem 33066
Description: Other elements of a scalar restriction. (Contributed by Thierry Arnoux, 6-Sep-2018.) (Revised by AV, 31-Oct-2024.)
Hypotheses
Ref Expression
resvlem.r 𝑅 = (π‘Š β†Ύv 𝐴)
resvlem.e 𝐢 = (πΈβ€˜π‘Š)
resvlem.f 𝐸 = Slot (πΈβ€˜ndx)
resvlem.n (πΈβ€˜ndx) β‰  (Scalarβ€˜ndx)
Assertion
Ref Expression
resvlem (𝐴 ∈ 𝑉 β†’ 𝐢 = (πΈβ€˜π‘…))

Proof of Theorem resvlem
StepHypRef Expression
1 resvlem.e . 2 𝐢 = (πΈβ€˜π‘Š)
2 resvlem.r . . . . . . 7 𝑅 = (π‘Š β†Ύv 𝐴)
3 eqid 2728 . . . . . . 7 (Scalarβ€˜π‘Š) = (Scalarβ€˜π‘Š)
4 eqid 2728 . . . . . . 7 (Baseβ€˜(Scalarβ€˜π‘Š)) = (Baseβ€˜(Scalarβ€˜π‘Š))
52, 3, 4resvid2 33063 . . . . . 6 (((Baseβ€˜(Scalarβ€˜π‘Š)) βŠ† 𝐴 ∧ π‘Š ∈ V ∧ 𝐴 ∈ 𝑉) β†’ 𝑅 = π‘Š)
65fveq2d 6906 . . . . 5 (((Baseβ€˜(Scalarβ€˜π‘Š)) βŠ† 𝐴 ∧ π‘Š ∈ V ∧ 𝐴 ∈ 𝑉) β†’ (πΈβ€˜π‘…) = (πΈβ€˜π‘Š))
763expib 1119 . . . 4 ((Baseβ€˜(Scalarβ€˜π‘Š)) βŠ† 𝐴 β†’ ((π‘Š ∈ V ∧ 𝐴 ∈ 𝑉) β†’ (πΈβ€˜π‘…) = (πΈβ€˜π‘Š)))
82, 3, 4resvval2 33064 . . . . . . 7 ((Β¬ (Baseβ€˜(Scalarβ€˜π‘Š)) βŠ† 𝐴 ∧ π‘Š ∈ V ∧ 𝐴 ∈ 𝑉) β†’ 𝑅 = (π‘Š sSet ⟨(Scalarβ€˜ndx), ((Scalarβ€˜π‘Š) β†Ύs 𝐴)⟩))
98fveq2d 6906 . . . . . 6 ((Β¬ (Baseβ€˜(Scalarβ€˜π‘Š)) βŠ† 𝐴 ∧ π‘Š ∈ V ∧ 𝐴 ∈ 𝑉) β†’ (πΈβ€˜π‘…) = (πΈβ€˜(π‘Š sSet ⟨(Scalarβ€˜ndx), ((Scalarβ€˜π‘Š) β†Ύs 𝐴)⟩)))
10 resvlem.f . . . . . . 7 𝐸 = Slot (πΈβ€˜ndx)
11 resvlem.n . . . . . . 7 (πΈβ€˜ndx) β‰  (Scalarβ€˜ndx)
1210, 11setsnid 17185 . . . . . 6 (πΈβ€˜π‘Š) = (πΈβ€˜(π‘Š sSet ⟨(Scalarβ€˜ndx), ((Scalarβ€˜π‘Š) β†Ύs 𝐴)⟩))
139, 12eqtr4di 2786 . . . . 5 ((Β¬ (Baseβ€˜(Scalarβ€˜π‘Š)) βŠ† 𝐴 ∧ π‘Š ∈ V ∧ 𝐴 ∈ 𝑉) β†’ (πΈβ€˜π‘…) = (πΈβ€˜π‘Š))
14133expib 1119 . . . 4 (Β¬ (Baseβ€˜(Scalarβ€˜π‘Š)) βŠ† 𝐴 β†’ ((π‘Š ∈ V ∧ 𝐴 ∈ 𝑉) β†’ (πΈβ€˜π‘…) = (πΈβ€˜π‘Š)))
157, 14pm2.61i 182 . . 3 ((π‘Š ∈ V ∧ 𝐴 ∈ 𝑉) β†’ (πΈβ€˜π‘…) = (πΈβ€˜π‘Š))
1610str0 17165 . . . . . . 7 βˆ… = (πΈβ€˜βˆ…)
1716eqcomi 2737 . . . . . 6 (πΈβ€˜βˆ…) = βˆ…
18 reldmresv 33061 . . . . . 6 Rel dom β†Ύv
1917, 2, 18oveqprc 17168 . . . . 5 (Β¬ π‘Š ∈ V β†’ (πΈβ€˜π‘Š) = (πΈβ€˜π‘…))
2019eqcomd 2734 . . . 4 (Β¬ π‘Š ∈ V β†’ (πΈβ€˜π‘…) = (πΈβ€˜π‘Š))
2120adantr 479 . . 3 ((Β¬ π‘Š ∈ V ∧ 𝐴 ∈ 𝑉) β†’ (πΈβ€˜π‘…) = (πΈβ€˜π‘Š))
2215, 21pm2.61ian 810 . 2 (𝐴 ∈ 𝑉 β†’ (πΈβ€˜π‘…) = (πΈβ€˜π‘Š))
231, 22eqtr4id 2787 1 (𝐴 ∈ 𝑉 β†’ 𝐢 = (πΈβ€˜π‘…))
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098   β‰  wne 2937  Vcvv 3473   βŠ† wss 3949  βˆ…c0 4326  βŸ¨cop 4638  β€˜cfv 6553  (class class class)co 7426   sSet csts 17139  Slot cslot 17157  ndxcnx 17169  Basecbs 17187   β†Ύs cress 17216  Scalarcsca 17243   β†Ύv cresv 33059
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7746
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-res 5694  df-iota 6505  df-fun 6555  df-fv 6561  df-ov 7429  df-oprab 7430  df-mpo 7431  df-sets 17140  df-slot 17158  df-resv 33060
This theorem is referenced by:  resvbas  33068  resvplusg  33070  resvvsca  33072  resvmulr  33074
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