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Theorem msrf 34200
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 5426 . . . . 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 5272 . . . 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 5272 . . 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 2733 . . . 4 (mVarsβ€˜π‘‡) = (mVarsβ€˜π‘‡)
5 mpstssv.p . . . 4 𝑃 = (mPreStβ€˜π‘‡)
6 msrf.r . . . 4 𝑅 = (mStRedβ€˜π‘‡)
74, 5, 6msrfval 34195 . . 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 6648 . 2 𝑅 Fn 𝑃
95mpst123 34198 . . . . . 6 (𝑠 ∈ 𝑃 β†’ 𝑠 = ⟨(1st β€˜(1st β€˜π‘ )), (2nd β€˜(1st β€˜π‘ )), (2nd β€˜π‘ )⟩)
109fveq2d 6850 . . . . 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 2835 . . . . . 6 (𝑠 ∈ 𝑃 β†’ ⟨(1st β€˜(1st β€˜π‘ )), (2nd β€˜(1st β€˜π‘ )), (2nd β€˜π‘ )⟩ ∈ 𝑃)
13 eqid 2733 . . . . . . 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 34196 . . . . . 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 2773 . . . 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 4192 . . . . . . 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 2733 . . . . . . . . . . 11 (mDVβ€˜π‘‡) = (mDVβ€˜π‘‡)
19 eqid 2733 . . . . . . . . . . 11 (mExβ€˜π‘‡) = (mExβ€˜π‘‡)
2018, 19, 5elmpst 34194 . . . . . . . . . 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 1143 . . . . . . . 8 (𝑠 ∈ 𝑃 β†’ ((1st β€˜(1st β€˜π‘ )) βŠ† (mDVβ€˜π‘‡) ∧ β—‘(1st β€˜(1st β€˜π‘ )) = (1st β€˜(1st β€˜π‘ ))))
2322simpld 496 . . . . . . 7 (𝑠 ∈ 𝑃 β†’ (1st β€˜(1st β€˜π‘ )) βŠ† (mDVβ€˜π‘‡))
2417, 23sstrid 3959 . . . . . 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 6101 . . . . . . 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 497 . . . . . . . 8 (𝑠 ∈ 𝑃 β†’ β—‘(1st β€˜(1st β€˜π‘ )) = (1st β€˜(1st β€˜π‘ )))
27 cnvxp 6113 . . . . . . . . 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 4177 . . . . . . 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 2785 . . . . . 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 513 . . . . 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 1144 . . . . 5 (𝑠 ∈ 𝑃 β†’ ((2nd β€˜(1st β€˜π‘ )) βŠ† (mExβ€˜π‘‡) ∧ (2nd β€˜(1st β€˜π‘ )) ∈ Fin))
3321simp3d 1145 . . . . 5 (𝑠 ∈ 𝑃 β†’ (2nd β€˜π‘ ) ∈ (mExβ€˜π‘‡))
3418, 19, 5elmpst 34194 . . . . 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 1344 . . . 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 2834 . . 3 (𝑠 ∈ 𝑃 β†’ (π‘…β€˜π‘ ) ∈ 𝑃)
3736rgen 3063 . 2 βˆ€π‘  ∈ 𝑃 (π‘…β€˜π‘ ) ∈ 𝑃
38 ffnfv 7070 . 2 (𝑅:π‘ƒβŸΆπ‘ƒ ↔ (𝑅 Fn 𝑃 ∧ βˆ€π‘  ∈ 𝑃 (π‘…β€˜π‘ ) ∈ 𝑃))
398, 37, 38mpbir2an 710 1 𝑅:π‘ƒβŸΆπ‘ƒ
Colors of variables: wff setvar class
Syntax hints:   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  β¦‹csb 3859   βˆͺ cun 3912   ∩ cin 3913   βŠ† wss 3914  {csn 4590  βŸ¨cotp 4598  βˆͺ cuni 4869   Γ— cxp 5635  β—‘ccnv 5636   β€œ cima 5640   Fn wfn 6495  βŸΆwf 6496  β€˜cfv 6500  1st c1st 7923  2nd c2nd 7924  Fincfn 8889  mExcmex 34125  mDVcmdv 34126  mVarscmvrs 34127  mPreStcmpst 34131  mStRedcmsr 34132
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-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
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-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-ot 4599  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-1st 7925  df-2nd 7926  df-mpst 34151  df-msr 34152
This theorem is referenced by:  msrrcl  34201  msrid  34203  msrfo  34204  mstapst  34205  elmsta  34206  elmthm  34234  mthmsta  34236  mthmblem  34238
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