Users' Mathboxes Mathbox for Mario Carneiro < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  msrval Structured version   Visualization version   GIF version

Theorem msrval 34196
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 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 (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 6861 . . 3 ((⟨𝐷, 𝐻, 𝐴⟩ ∈ 𝑃 ∧ 𝑠 = ⟨𝐷, 𝐻, 𝐴⟩) β†’ (2nd β€˜(1st β€˜π‘ )) ∈ V)
7 fvexd 6861 . . . 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 6850 . . . . . . . 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 6850 . . . . . . 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 34194 . . . . . . . . . . . 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 6859 . . . . . . . . . 10 (mDVβ€˜π‘‡) ∈ V
1817ssex 5282 . . . . . . . . 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 7939 . . . . . . . 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 6860 . . . . . . . . . 10 𝑉 ∈ V
29 imaexg 7856 . . . . . . . . . 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 7682 . . . . . . . 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 6850 . . . . . . . . . . . . . 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 7940 . . . . . . . . . . . . . . 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 6850 . . . . . . . . . . . . . . 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 7941 . . . . . . . . . . . . . . . 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 4602 . . . . . . . . . . . . 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 4128 . . . . . . . . . . . 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 6017 . . . . . . . . . . 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 4883 . . . . . . . . . 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 5669 . . . . . . 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 3897 . . . . . 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 4177 . . . . 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 4849 . . . 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 3897 . . 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 3897 . 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 5426 . . 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 6959 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 3447  β¦‹csb 3859   βˆͺ cun 3912   ∩ cin 3913   βŠ† wss 3914  {csn 4590  βŸ¨cotp 4598  βˆͺ cuni 4869   ↦ cmpt 5192   Γ— cxp 5635  β—‘ccnv 5636   β€œ cima 5640  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:  msrf  34200  msrid  34203  elmsta  34206  mthmpps  34240
  Copyright terms: Public domain W3C validator