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Theorem msrf 34528
Description: The reduct of a pre-statement is a pre-statement. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
mpstssv.p 𝑃 = (mPreStβ€˜π‘‡)
msrf.r 𝑅 = (mStRedβ€˜π‘‡)
Assertion
Ref Expression
msrf 𝑅:π‘ƒβŸΆπ‘ƒ

Proof of Theorem msrf
Dummy variables β„Ž π‘Ž 𝑠 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 otex 5465 . . . . 5 ⟨((1st β€˜(1st β€˜π‘ )) ∩ ⦋βˆͺ ((mVarsβ€˜π‘‡) β€œ (β„Ž βˆͺ {π‘Ž})) / π‘§β¦Œ(𝑧 Γ— 𝑧)), β„Ž, π‘ŽβŸ© ∈ V
21csbex 5311 . . . 4 ⦋(2nd β€˜π‘ ) / π‘Žβ¦ŒβŸ¨((1st β€˜(1st β€˜π‘ )) ∩ ⦋βˆͺ ((mVarsβ€˜π‘‡) β€œ (β„Ž βˆͺ {π‘Ž})) / π‘§β¦Œ(𝑧 Γ— 𝑧)), β„Ž, π‘ŽβŸ© ∈ V
32csbex 5311 . . 3 ⦋(2nd β€˜(1st β€˜π‘ )) / β„Žβ¦Œβ¦‹(2nd β€˜π‘ ) / π‘Žβ¦ŒβŸ¨((1st β€˜(1st β€˜π‘ )) ∩ ⦋βˆͺ ((mVarsβ€˜π‘‡) β€œ (β„Ž βˆͺ {π‘Ž})) / π‘§β¦Œ(𝑧 Γ— 𝑧)), β„Ž, π‘ŽβŸ© ∈ V
4 eqid 2732 . . . 4 (mVarsβ€˜π‘‡) = (mVarsβ€˜π‘‡)
5 mpstssv.p . . . 4 𝑃 = (mPreStβ€˜π‘‡)
6 msrf.r . . . 4 𝑅 = (mStRedβ€˜π‘‡)
74, 5, 6msrfval 34523 . . 3 𝑅 = (𝑠 ∈ 𝑃 ↦ ⦋(2nd β€˜(1st β€˜π‘ )) / β„Žβ¦Œβ¦‹(2nd β€˜π‘ ) / π‘Žβ¦ŒβŸ¨((1st β€˜(1st β€˜π‘ )) ∩ ⦋βˆͺ ((mVarsβ€˜π‘‡) β€œ (β„Ž βˆͺ {π‘Ž})) / π‘§β¦Œ(𝑧 Γ— 𝑧)), β„Ž, π‘ŽβŸ©)
83, 7fnmpti 6693 . 2 𝑅 Fn 𝑃
95mpst123 34526 . . . . . 6 (𝑠 ∈ 𝑃 β†’ 𝑠 = ⟨(1st β€˜(1st β€˜π‘ )), (2nd β€˜(1st β€˜π‘ )), (2nd β€˜π‘ )⟩)
109fveq2d 6895 . . . . 5 (𝑠 ∈ 𝑃 β†’ (π‘…β€˜π‘ ) = (π‘…β€˜βŸ¨(1st β€˜(1st β€˜π‘ )), (2nd β€˜(1st β€˜π‘ )), (2nd β€˜π‘ )⟩))
11 id 22 . . . . . . 7 (𝑠 ∈ 𝑃 β†’ 𝑠 ∈ 𝑃)
129, 11eqeltrrd 2834 . . . . . 6 (𝑠 ∈ 𝑃 β†’ ⟨(1st β€˜(1st β€˜π‘ )), (2nd β€˜(1st β€˜π‘ )), (2nd β€˜π‘ )⟩ ∈ 𝑃)
13 eqid 2732 . . . . . . 7 βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})) = βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )}))
144, 5, 6, 13msrval 34524 . . . . . 6 (⟨(1st β€˜(1st β€˜π‘ )), (2nd β€˜(1st β€˜π‘ )), (2nd β€˜π‘ )⟩ ∈ 𝑃 β†’ (π‘…β€˜βŸ¨(1st β€˜(1st β€˜π‘ )), (2nd β€˜(1st β€˜π‘ )), (2nd β€˜π‘ )⟩) = ⟨((1st β€˜(1st β€˜π‘ )) ∩ (βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})) Γ— βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})))), (2nd β€˜(1st β€˜π‘ )), (2nd β€˜π‘ )⟩)
1512, 14syl 17 . . . . 5 (𝑠 ∈ 𝑃 β†’ (π‘…β€˜βŸ¨(1st β€˜(1st β€˜π‘ )), (2nd β€˜(1st β€˜π‘ )), (2nd β€˜π‘ )⟩) = ⟨((1st β€˜(1st β€˜π‘ )) ∩ (βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})) Γ— βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})))), (2nd β€˜(1st β€˜π‘ )), (2nd β€˜π‘ )⟩)
1610, 15eqtrd 2772 . . . 4 (𝑠 ∈ 𝑃 β†’ (π‘…β€˜π‘ ) = ⟨((1st β€˜(1st β€˜π‘ )) ∩ (βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})) Γ— βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})))), (2nd β€˜(1st β€˜π‘ )), (2nd β€˜π‘ )⟩)
17 inss1 4228 . . . . . . 7 ((1st β€˜(1st β€˜π‘ )) ∩ (βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})) Γ— βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})))) βŠ† (1st β€˜(1st β€˜π‘ ))
18 eqid 2732 . . . . . . . . . . 11 (mDVβ€˜π‘‡) = (mDVβ€˜π‘‡)
19 eqid 2732 . . . . . . . . . . 11 (mExβ€˜π‘‡) = (mExβ€˜π‘‡)
2018, 19, 5elmpst 34522 . . . . . . . . . 10 (⟨(1st β€˜(1st β€˜π‘ )), (2nd β€˜(1st β€˜π‘ )), (2nd β€˜π‘ )⟩ ∈ 𝑃 ↔ (((1st β€˜(1st β€˜π‘ )) βŠ† (mDVβ€˜π‘‡) ∧ β—‘(1st β€˜(1st β€˜π‘ )) = (1st β€˜(1st β€˜π‘ ))) ∧ ((2nd β€˜(1st β€˜π‘ )) βŠ† (mExβ€˜π‘‡) ∧ (2nd β€˜(1st β€˜π‘ )) ∈ Fin) ∧ (2nd β€˜π‘ ) ∈ (mExβ€˜π‘‡)))
2112, 20sylib 217 . . . . . . . . 9 (𝑠 ∈ 𝑃 β†’ (((1st β€˜(1st β€˜π‘ )) βŠ† (mDVβ€˜π‘‡) ∧ β—‘(1st β€˜(1st β€˜π‘ )) = (1st β€˜(1st β€˜π‘ ))) ∧ ((2nd β€˜(1st β€˜π‘ )) βŠ† (mExβ€˜π‘‡) ∧ (2nd β€˜(1st β€˜π‘ )) ∈ Fin) ∧ (2nd β€˜π‘ ) ∈ (mExβ€˜π‘‡)))
2221simp1d 1142 . . . . . . . 8 (𝑠 ∈ 𝑃 β†’ ((1st β€˜(1st β€˜π‘ )) βŠ† (mDVβ€˜π‘‡) ∧ β—‘(1st β€˜(1st β€˜π‘ )) = (1st β€˜(1st β€˜π‘ ))))
2322simpld 495 . . . . . . 7 (𝑠 ∈ 𝑃 β†’ (1st β€˜(1st β€˜π‘ )) βŠ† (mDVβ€˜π‘‡))
2417, 23sstrid 3993 . . . . . 6 (𝑠 ∈ 𝑃 β†’ ((1st β€˜(1st β€˜π‘ )) ∩ (βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})) Γ— βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})))) βŠ† (mDVβ€˜π‘‡))
25 cnvin 6144 . . . . . . 7 β—‘((1st β€˜(1st β€˜π‘ )) ∩ (βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})) Γ— βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})))) = (β—‘(1st β€˜(1st β€˜π‘ )) ∩ β—‘(βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})) Γ— βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )}))))
2622simprd 496 . . . . . . . 8 (𝑠 ∈ 𝑃 β†’ β—‘(1st β€˜(1st β€˜π‘ )) = (1st β€˜(1st β€˜π‘ )))
27 cnvxp 6156 . . . . . . . . 9 β—‘(βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})) Γ— βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )}))) = (βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})) Γ— βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})))
2827a1i 11 . . . . . . . 8 (𝑠 ∈ 𝑃 β†’ β—‘(βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})) Γ— βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )}))) = (βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})) Γ— βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )}))))
2926, 28ineq12d 4213 . . . . . . 7 (𝑠 ∈ 𝑃 β†’ (β—‘(1st β€˜(1st β€˜π‘ )) ∩ β—‘(βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})) Γ— βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})))) = ((1st β€˜(1st β€˜π‘ )) ∩ (βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})) Γ— βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})))))
3025, 29eqtrid 2784 . . . . . 6 (𝑠 ∈ 𝑃 β†’ β—‘((1st β€˜(1st β€˜π‘ )) ∩ (βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})) Γ— βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})))) = ((1st β€˜(1st β€˜π‘ )) ∩ (βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})) Γ— βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})))))
3124, 30jca 512 . . . . 5 (𝑠 ∈ 𝑃 β†’ (((1st β€˜(1st β€˜π‘ )) ∩ (βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})) Γ— βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})))) βŠ† (mDVβ€˜π‘‡) ∧ β—‘((1st β€˜(1st β€˜π‘ )) ∩ (βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})) Γ— βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})))) = ((1st β€˜(1st β€˜π‘ )) ∩ (βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})) Γ— βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )}))))))
3221simp2d 1143 . . . . 5 (𝑠 ∈ 𝑃 β†’ ((2nd β€˜(1st β€˜π‘ )) βŠ† (mExβ€˜π‘‡) ∧ (2nd β€˜(1st β€˜π‘ )) ∈ Fin))
3321simp3d 1144 . . . . 5 (𝑠 ∈ 𝑃 β†’ (2nd β€˜π‘ ) ∈ (mExβ€˜π‘‡))
3418, 19, 5elmpst 34522 . . . . 5 (⟨((1st β€˜(1st β€˜π‘ )) ∩ (βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})) Γ— βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})))), (2nd β€˜(1st β€˜π‘ )), (2nd β€˜π‘ )⟩ ∈ 𝑃 ↔ ((((1st β€˜(1st β€˜π‘ )) ∩ (βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})) Γ— βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})))) βŠ† (mDVβ€˜π‘‡) ∧ β—‘((1st β€˜(1st β€˜π‘ )) ∩ (βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})) Γ— βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})))) = ((1st β€˜(1st β€˜π‘ )) ∩ (βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})) Γ— βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )}))))) ∧ ((2nd β€˜(1st β€˜π‘ )) βŠ† (mExβ€˜π‘‡) ∧ (2nd β€˜(1st β€˜π‘ )) ∈ Fin) ∧ (2nd β€˜π‘ ) ∈ (mExβ€˜π‘‡)))
3531, 32, 33, 34syl3anbrc 1343 . . . 4 (𝑠 ∈ 𝑃 β†’ ⟨((1st β€˜(1st β€˜π‘ )) ∩ (βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})) Γ— βˆͺ ((mVarsβ€˜π‘‡) β€œ ((2nd β€˜(1st β€˜π‘ )) βˆͺ {(2nd β€˜π‘ )})))), (2nd β€˜(1st β€˜π‘ )), (2nd β€˜π‘ )⟩ ∈ 𝑃)
3616, 35eqeltrd 2833 . . 3 (𝑠 ∈ 𝑃 β†’ (π‘…β€˜π‘ ) ∈ 𝑃)
3736rgen 3063 . 2 βˆ€π‘  ∈ 𝑃 (π‘…β€˜π‘ ) ∈ 𝑃
38 ffnfv 7117 . 2 (𝑅:π‘ƒβŸΆπ‘ƒ ↔ (𝑅 Fn 𝑃 ∧ βˆ€π‘  ∈ 𝑃 (π‘…β€˜π‘ ) ∈ 𝑃))
398, 37, 38mpbir2an 709 1 𝑅:π‘ƒβŸΆπ‘ƒ
Colors of variables: wff setvar class
Syntax hints:   ∧ wa 396   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  β¦‹csb 3893   βˆͺ cun 3946   ∩ cin 3947   βŠ† wss 3948  {csn 4628  βŸ¨cotp 4636  βˆͺ cuni 4908   Γ— cxp 5674  β—‘ccnv 5675   β€œ cima 5679   Fn wfn 6538  βŸΆwf 6539  β€˜cfv 6543  1st c1st 7972  2nd c2nd 7973  Fincfn 8938  mExcmex 34453  mDVcmdv 34454  mVarscmvrs 34455  mPreStcmpst 34459  mStRedcmsr 34460
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-ot 4637  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-1st 7974  df-2nd 7975  df-mpst 34479  df-msr 34480
This theorem is referenced by:  msrrcl  34529  msrid  34531  msrfo  34532  mstapst  34533  elmsta  34534  elmthm  34562  mthmsta  34564  mthmblem  34566
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