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Theorem msrval 34529
Description: Value of the reduct of a pre-statement. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypotheses
Ref Expression
msrfval.v 𝑉 = (mVarsβ€˜π‘‡)
msrfval.p 𝑃 = (mPreStβ€˜π‘‡)
msrfval.r 𝑅 = (mStRedβ€˜π‘‡)
msrval.z 𝑍 = βˆͺ (𝑉 β€œ (𝐻 βˆͺ {𝐴}))
Assertion
Ref Expression
msrval (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 β†’ (π‘…β€˜βŸ¨π·, 𝐻, 𝐴⟩) = ⟨(𝐷 ∩ (𝑍 Γ— 𝑍)), 𝐻, 𝐴⟩)

Proof of Theorem msrval
Dummy variables β„Ž π‘Ž 𝑠 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 msrfval.v . . . 4 𝑉 = (mVarsβ€˜π‘‡)
2 msrfval.p . . . 4 𝑃 = (mPreStβ€˜π‘‡)
3 msrfval.r . . . 4 𝑅 = (mStRedβ€˜π‘‡)
41, 2, 3msrfval 34528 . . 3 𝑅 = (𝑠 ∈ 𝑃 ↦ ⦋(2nd β€˜(1st β€˜π‘ )) / β„Žβ¦Œβ¦‹(2nd β€˜π‘ ) / π‘Žβ¦ŒβŸ¨((1st β€˜(1st β€˜π‘ )) ∩ ⦋βˆͺ (𝑉 β€œ (β„Ž βˆͺ {π‘Ž})) / π‘§β¦Œ(𝑧 Γ— 𝑧)), β„Ž, π‘ŽβŸ©)
54a1i 11 . 2 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 β†’ 𝑅 = (𝑠 ∈ 𝑃 ↦ ⦋(2nd β€˜(1st β€˜π‘ )) / β„Žβ¦Œβ¦‹(2nd β€˜π‘ ) / π‘Žβ¦ŒβŸ¨((1st β€˜(1st β€˜π‘ )) ∩ ⦋βˆͺ (𝑉 β€œ (β„Ž βˆͺ {π‘Ž})) / π‘§β¦Œ(𝑧 Γ— 𝑧)), β„Ž, π‘ŽβŸ©))
6 fvexd 6907 . . 3 ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) β†’ (2nd β€˜(1st β€˜π‘ )) ∈ V)
7 fvexd 6907 . . . 4 (((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ β„Ž = (2nd β€˜(1st β€˜π‘ ))) β†’ (2nd β€˜π‘ ) ∈ V)
8 simpllr 775 . . . . . . . . 9 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ β„Ž = (2nd β€˜(1st β€˜π‘ ))) ∧ π‘Ž = (2nd β€˜π‘ )) β†’ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩)
98fveq2d 6896 . . . . . . . 8 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ β„Ž = (2nd β€˜(1st β€˜π‘ ))) ∧ π‘Ž = (2nd β€˜π‘ )) β†’ (1st β€˜π‘ ) = (1st β€˜βŸ¨π·, 𝐻, 𝐴⟩))
109fveq2d 6896 . . . . . . 7 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ β„Ž = (2nd β€˜(1st β€˜π‘ ))) ∧ π‘Ž = (2nd β€˜π‘ )) β†’ (1st β€˜(1st β€˜π‘ )) = (1st β€˜(1st β€˜βŸ¨π·, 𝐻, 𝐴⟩)))
11 eqid 2733 . . . . . . . . . . . . 13 (mDVβ€˜π‘‡) = (mDVβ€˜π‘‡)
12 eqid 2733 . . . . . . . . . . . . 13 (mExβ€˜π‘‡) = (mExβ€˜π‘‡)
1311, 12, 2elmpst 34527 . . . . . . . . . . . 12 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ↔ ((𝐷 βŠ† (mDVβ€˜π‘‡) ∧ ◑𝐷 = 𝐷) ∧ (𝐻 βŠ† (mExβ€˜π‘‡) ∧ 𝐻 ∈ Fin) ∧ 𝐴 ∈ (mExβ€˜π‘‡)))
1413simp1bi 1146 . . . . . . . . . . 11 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 β†’ (𝐷 βŠ† (mDVβ€˜π‘‡) ∧ ◑𝐷 = 𝐷))
1514simpld 496 . . . . . . . . . 10 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 β†’ 𝐷 βŠ† (mDVβ€˜π‘‡))
1615ad3antrrr 729 . . . . . . . . 9 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ β„Ž = (2nd β€˜(1st β€˜π‘ ))) ∧ π‘Ž = (2nd β€˜π‘ )) β†’ 𝐷 βŠ† (mDVβ€˜π‘‡))
17 fvex 6905 . . . . . . . . . 10 (mDVβ€˜π‘‡) ∈ V
1817ssex 5322 . . . . . . . . 9 (𝐷 βŠ† (mDVβ€˜π‘‡) β†’ 𝐷 ∈ V)
1916, 18syl 17 . . . . . . . 8 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ β„Ž = (2nd β€˜(1st β€˜π‘ ))) ∧ π‘Ž = (2nd β€˜π‘ )) β†’ 𝐷 ∈ V)
2013simp2bi 1147 . . . . . . . . . 10 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 β†’ (𝐻 βŠ† (mExβ€˜π‘‡) ∧ 𝐻 ∈ Fin))
2120simprd 497 . . . . . . . . 9 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 β†’ 𝐻 ∈ Fin)
2221ad3antrrr 729 . . . . . . . 8 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ β„Ž = (2nd β€˜(1st β€˜π‘ ))) ∧ π‘Ž = (2nd β€˜π‘ )) β†’ 𝐻 ∈ Fin)
2313simp3bi 1148 . . . . . . . . 9 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 β†’ 𝐴 ∈ (mExβ€˜π‘‡))
2423ad3antrrr 729 . . . . . . . 8 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ β„Ž = (2nd β€˜(1st β€˜π‘ ))) ∧ π‘Ž = (2nd β€˜π‘ )) β†’ 𝐴 ∈ (mExβ€˜π‘‡))
25 ot1stg 7989 . . . . . . . 8 ((𝐷 ∈ V ∧ 𝐻 ∈ Fin ∧ 𝐴 ∈ (mExβ€˜π‘‡)) β†’ (1st β€˜(1st β€˜βŸ¨π·, 𝐻, 𝐴⟩)) = 𝐷)
2619, 22, 24, 25syl3anc 1372 . . . . . . 7 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ β„Ž = (2nd β€˜(1st β€˜π‘ ))) ∧ π‘Ž = (2nd β€˜π‘ )) β†’ (1st β€˜(1st β€˜βŸ¨π·, 𝐻, 𝐴⟩)) = 𝐷)
2710, 26eqtrd 2773 . . . . . 6 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ β„Ž = (2nd β€˜(1st β€˜π‘ ))) ∧ π‘Ž = (2nd β€˜π‘ )) β†’ (1st β€˜(1st β€˜π‘ )) = 𝐷)
281fvexi 6906 . . . . . . . . . 10 𝑉 ∈ V
29 imaexg 7906 . . . . . . . . . 10 (𝑉 ∈ V β†’ (𝑉 β€œ (β„Ž βˆͺ {π‘Ž})) ∈ V)
3028, 29ax-mp 5 . . . . . . . . 9 (𝑉 β€œ (β„Ž βˆͺ {π‘Ž})) ∈ V
3130uniex 7731 . . . . . . . 8 βˆͺ (𝑉 β€œ (β„Ž βˆͺ {π‘Ž})) ∈ V
3231a1i 11 . . . . . . 7 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ β„Ž = (2nd β€˜(1st β€˜π‘ ))) ∧ π‘Ž = (2nd β€˜π‘ )) β†’ βˆͺ (𝑉 β€œ (β„Ž βˆͺ {π‘Ž})) ∈ V)
33 id 22 . . . . . . . . 9 (𝑧 = βˆͺ (𝑉 β€œ (β„Ž βˆͺ {π‘Ž})) β†’ 𝑧 = βˆͺ (𝑉 β€œ (β„Ž βˆͺ {π‘Ž})))
34 simplr 768 . . . . . . . . . . . . . 14 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ β„Ž = (2nd β€˜(1st β€˜π‘ ))) ∧ π‘Ž = (2nd β€˜π‘ )) β†’ β„Ž = (2nd β€˜(1st β€˜π‘ )))
359fveq2d 6896 . . . . . . . . . . . . . 14 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ β„Ž = (2nd β€˜(1st β€˜π‘ ))) ∧ π‘Ž = (2nd β€˜π‘ )) β†’ (2nd β€˜(1st β€˜π‘ )) = (2nd β€˜(1st β€˜βŸ¨π·, 𝐻, 𝐴⟩)))
36 ot2ndg 7990 . . . . . . . . . . . . . . 15 ((𝐷 ∈ V ∧ 𝐻 ∈ Fin ∧ 𝐴 ∈ (mExβ€˜π‘‡)) β†’ (2nd β€˜(1st β€˜βŸ¨π·, 𝐻, 𝐴⟩)) = 𝐻)
3719, 22, 24, 36syl3anc 1372 . . . . . . . . . . . . . 14 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ β„Ž = (2nd β€˜(1st β€˜π‘ ))) ∧ π‘Ž = (2nd β€˜π‘ )) β†’ (2nd β€˜(1st β€˜βŸ¨π·, 𝐻, 𝐴⟩)) = 𝐻)
3834, 35, 373eqtrd 2777 . . . . . . . . . . . . 13 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ β„Ž = (2nd β€˜(1st β€˜π‘ ))) ∧ π‘Ž = (2nd β€˜π‘ )) β†’ β„Ž = 𝐻)
39 simpr 486 . . . . . . . . . . . . . . 15 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ β„Ž = (2nd β€˜(1st β€˜π‘ ))) ∧ π‘Ž = (2nd β€˜π‘ )) β†’ π‘Ž = (2nd β€˜π‘ ))
408fveq2d 6896 . . . . . . . . . . . . . . 15 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ β„Ž = (2nd β€˜(1st β€˜π‘ ))) ∧ π‘Ž = (2nd β€˜π‘ )) β†’ (2nd β€˜π‘ ) = (2nd β€˜βŸ¨π·, 𝐻, 𝐴⟩))
41 ot3rdg 7991 . . . . . . . . . . . . . . . 16 (𝐴 ∈ (mExβ€˜π‘‡) β†’ (2nd β€˜βŸ¨π·, 𝐻, 𝐴⟩) = 𝐴)
4224, 41syl 17 . . . . . . . . . . . . . . 15 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ β„Ž = (2nd β€˜(1st β€˜π‘ ))) ∧ π‘Ž = (2nd β€˜π‘ )) β†’ (2nd β€˜βŸ¨π·, 𝐻, 𝐴⟩) = 𝐴)
4339, 40, 423eqtrd 2777 . . . . . . . . . . . . . 14 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ β„Ž = (2nd β€˜(1st β€˜π‘ ))) ∧ π‘Ž = (2nd β€˜π‘ )) β†’ π‘Ž = 𝐴)
4443sneqd 4641 . . . . . . . . . . . . 13 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ β„Ž = (2nd β€˜(1st β€˜π‘ ))) ∧ π‘Ž = (2nd β€˜π‘ )) β†’ {π‘Ž} = {𝐴})
4538, 44uneq12d 4165 . . . . . . . . . . . 12 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ β„Ž = (2nd β€˜(1st β€˜π‘ ))) ∧ π‘Ž = (2nd β€˜π‘ )) β†’ (β„Ž βˆͺ {π‘Ž}) = (𝐻 βˆͺ {𝐴}))
4645imaeq2d 6060 . . . . . . . . . . 11 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ β„Ž = (2nd β€˜(1st β€˜π‘ ))) ∧ π‘Ž = (2nd β€˜π‘ )) β†’ (𝑉 β€œ (β„Ž βˆͺ {π‘Ž})) = (𝑉 β€œ (𝐻 βˆͺ {𝐴})))
4746unieqd 4923 . . . . . . . . . 10 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ β„Ž = (2nd β€˜(1st β€˜π‘ ))) ∧ π‘Ž = (2nd β€˜π‘ )) β†’ βˆͺ (𝑉 β€œ (β„Ž βˆͺ {π‘Ž})) = βˆͺ (𝑉 β€œ (𝐻 βˆͺ {𝐴})))
48 msrval.z . . . . . . . . . 10 𝑍 = βˆͺ (𝑉 β€œ (𝐻 βˆͺ {𝐴}))
4947, 48eqtr4di 2791 . . . . . . . . 9 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ β„Ž = (2nd β€˜(1st β€˜π‘ ))) ∧ π‘Ž = (2nd β€˜π‘ )) β†’ βˆͺ (𝑉 β€œ (β„Ž βˆͺ {π‘Ž})) = 𝑍)
5033, 49sylan9eqr 2795 . . . . . . . 8 (((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ β„Ž = (2nd β€˜(1st β€˜π‘ ))) ∧ π‘Ž = (2nd β€˜π‘ )) ∧ 𝑧 = βˆͺ (𝑉 β€œ (β„Ž βˆͺ {π‘Ž}))) β†’ 𝑧 = 𝑍)
5150sqxpeqd 5709 . . . . . . 7 (((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ β„Ž = (2nd β€˜(1st β€˜π‘ ))) ∧ π‘Ž = (2nd β€˜π‘ )) ∧ 𝑧 = βˆͺ (𝑉 β€œ (β„Ž βˆͺ {π‘Ž}))) β†’ (𝑧 Γ— 𝑧) = (𝑍 Γ— 𝑍))
5232, 51csbied 3932 . . . . . 6 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ β„Ž = (2nd β€˜(1st β€˜π‘ ))) ∧ π‘Ž = (2nd β€˜π‘ )) β†’ ⦋βˆͺ (𝑉 β€œ (β„Ž βˆͺ {π‘Ž})) / π‘§β¦Œ(𝑧 Γ— 𝑧) = (𝑍 Γ— 𝑍))
5327, 52ineq12d 4214 . . . . 5 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ β„Ž = (2nd β€˜(1st β€˜π‘ ))) ∧ π‘Ž = (2nd β€˜π‘ )) β†’ ((1st β€˜(1st β€˜π‘ )) ∩ ⦋βˆͺ (𝑉 β€œ (β„Ž βˆͺ {π‘Ž})) / π‘§β¦Œ(𝑧 Γ— 𝑧)) = (𝐷 ∩ (𝑍 Γ— 𝑍)))
5453, 38, 43oteq123d 4889 . . . 4 ((((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ β„Ž = (2nd β€˜(1st β€˜π‘ ))) ∧ π‘Ž = (2nd β€˜π‘ )) β†’ ⟨((1st β€˜(1st β€˜π‘ )) ∩ ⦋βˆͺ (𝑉 β€œ (β„Ž βˆͺ {π‘Ž})) / π‘§β¦Œ(𝑧 Γ— 𝑧)), β„Ž, π‘ŽβŸ© = ⟨(𝐷 ∩ (𝑍 Γ— 𝑍)), 𝐻, 𝐴⟩)
557, 54csbied 3932 . . 3 (((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) ∧ β„Ž = (2nd β€˜(1st β€˜π‘ ))) β†’ ⦋(2nd β€˜π‘ ) / π‘Žβ¦ŒβŸ¨((1st β€˜(1st β€˜π‘ )) ∩ ⦋βˆͺ (𝑉 β€œ (β„Ž βˆͺ {π‘Ž})) / π‘§β¦Œ(𝑧 Γ— 𝑧)), β„Ž, π‘ŽβŸ© = ⟨(𝐷 ∩ (𝑍 Γ— 𝑍)), 𝐻, 𝐴⟩)
566, 55csbied 3932 . 2 ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) β†’ ⦋(2nd β€˜(1st β€˜π‘ )) / β„Žβ¦Œβ¦‹(2nd β€˜π‘ ) / π‘Žβ¦ŒβŸ¨((1st β€˜(1st β€˜π‘ )) ∩ ⦋βˆͺ (𝑉 β€œ (β„Ž βˆͺ {π‘Ž})) / π‘§β¦Œ(𝑧 Γ— 𝑧)), β„Ž, π‘ŽβŸ© = ⟨(𝐷 ∩ (𝑍 Γ— 𝑍)), 𝐻, 𝐴⟩)
57 id 22 . 2 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 β†’ ⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃)
58 otex 5466 . . 3 ⟨(𝐷 ∩ (𝑍 Γ— 𝑍)), 𝐻, 𝐴⟩ ∈ V
5958a1i 11 . 2 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 β†’ ⟨(𝐷 ∩ (𝑍 Γ— 𝑍)), 𝐻, 𝐴⟩ ∈ V)
605, 56, 57, 59fvmptd 7006 1 (⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 β†’ (π‘…β€˜βŸ¨π·, 𝐻, 𝐴⟩) = ⟨(𝐷 ∩ (𝑍 Γ— 𝑍)), 𝐻, 𝐴⟩)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542   ∈ wcel 2107  Vcvv 3475  β¦‹csb 3894   βˆͺ cun 3947   ∩ cin 3948   βŠ† wss 3949  {csn 4629  βŸ¨cotp 4637  βˆͺ cuni 4909   ↦ cmpt 5232   Γ— cxp 5675  β—‘ccnv 5676   β€œ cima 5680  β€˜cfv 6544  1st c1st 7973  2nd c2nd 7974  Fincfn 8939  mExcmex 34458  mDVcmdv 34459  mVarscmvrs 34460  mPreStcmpst 34464  mStRedcmsr 34465
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 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
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 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-ot 4638  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-1st 7975  df-2nd 7976  df-mpst 34484  df-msr 34485
This theorem is referenced by:  msrf  34533  msrid  34536  elmsta  34539  mthmpps  34573
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