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

Theorem msubco 34818
Description: The composition of two substitutions is a substitution. (Contributed by Mario Carneiro, 18-Jul-2016.)
Hypothesis
Ref Expression
msubco.s 𝑆 = (mSubstβ€˜π‘‡)
Assertion
Ref Expression
msubco ((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) β†’ (𝐹 ∘ 𝐺) ∈ ran 𝑆)

Proof of Theorem msubco
Dummy variables 𝑓 𝑔 β„Ž π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2730 . . . . 5 (mExβ€˜π‘‡) = (mExβ€˜π‘‡)
2 eqid 2730 . . . . 5 (mRSubstβ€˜π‘‡) = (mRSubstβ€˜π‘‡)
3 msubco.s . . . . 5 𝑆 = (mSubstβ€˜π‘‡)
41, 2, 3elmsubrn 34815 . . . 4 ran 𝑆 = ran (𝑓 ∈ ran (mRSubstβ€˜π‘‡) ↦ (π‘₯ ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘₯), (π‘“β€˜(2nd β€˜π‘₯))⟩))
54eleq2i 2823 . . 3 (𝐹 ∈ ran 𝑆 ↔ 𝐹 ∈ ran (𝑓 ∈ ran (mRSubstβ€˜π‘‡) ↦ (π‘₯ ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘₯), (π‘“β€˜(2nd β€˜π‘₯))⟩)))
6 eqid 2730 . . . 4 (𝑓 ∈ ran (mRSubstβ€˜π‘‡) ↦ (π‘₯ ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘₯), (π‘“β€˜(2nd β€˜π‘₯))⟩)) = (𝑓 ∈ ran (mRSubstβ€˜π‘‡) ↦ (π‘₯ ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘₯), (π‘“β€˜(2nd β€˜π‘₯))⟩))
7 fvex 6905 . . . . 5 (mExβ€˜π‘‡) ∈ V
87mptex 7228 . . . 4 (π‘₯ ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘₯), (π‘“β€˜(2nd β€˜π‘₯))⟩) ∈ V
96, 8elrnmpti 5960 . . 3 (𝐹 ∈ ran (𝑓 ∈ ran (mRSubstβ€˜π‘‡) ↦ (π‘₯ ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘₯), (π‘“β€˜(2nd β€˜π‘₯))⟩)) ↔ βˆƒπ‘“ ∈ ran (mRSubstβ€˜π‘‡)𝐹 = (π‘₯ ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘₯), (π‘“β€˜(2nd β€˜π‘₯))⟩))
105, 9bitri 274 . 2 (𝐹 ∈ ran 𝑆 ↔ βˆƒπ‘“ ∈ ran (mRSubstβ€˜π‘‡)𝐹 = (π‘₯ ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘₯), (π‘“β€˜(2nd β€˜π‘₯))⟩))
111, 2, 3elmsubrn 34815 . . . 4 ran 𝑆 = ran (𝑔 ∈ ran (mRSubstβ€˜π‘‡) ↦ (𝑦 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘¦), (π‘”β€˜(2nd β€˜π‘¦))⟩))
1211eleq2i 2823 . . 3 (𝐺 ∈ ran 𝑆 ↔ 𝐺 ∈ ran (𝑔 ∈ ran (mRSubstβ€˜π‘‡) ↦ (𝑦 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘¦), (π‘”β€˜(2nd β€˜π‘¦))⟩)))
13 eqid 2730 . . . 4 (𝑔 ∈ ran (mRSubstβ€˜π‘‡) ↦ (𝑦 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘¦), (π‘”β€˜(2nd β€˜π‘¦))⟩)) = (𝑔 ∈ ran (mRSubstβ€˜π‘‡) ↦ (𝑦 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘¦), (π‘”β€˜(2nd β€˜π‘¦))⟩))
147mptex 7228 . . . 4 (𝑦 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘¦), (π‘”β€˜(2nd β€˜π‘¦))⟩) ∈ V
1513, 14elrnmpti 5960 . . 3 (𝐺 ∈ ran (𝑔 ∈ ran (mRSubstβ€˜π‘‡) ↦ (𝑦 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘¦), (π‘”β€˜(2nd β€˜π‘¦))⟩)) ↔ βˆƒπ‘” ∈ ran (mRSubstβ€˜π‘‡)𝐺 = (𝑦 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘¦), (π‘”β€˜(2nd β€˜π‘¦))⟩))
1612, 15bitri 274 . 2 (𝐺 ∈ ran 𝑆 ↔ βˆƒπ‘” ∈ ran (mRSubstβ€˜π‘‡)𝐺 = (𝑦 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘¦), (π‘”β€˜(2nd β€˜π‘¦))⟩))
17 reeanv 3224 . . 3 (βˆƒπ‘“ ∈ ran (mRSubstβ€˜π‘‡)βˆƒπ‘” ∈ ran (mRSubstβ€˜π‘‡)(𝐹 = (π‘₯ ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘₯), (π‘“β€˜(2nd β€˜π‘₯))⟩) ∧ 𝐺 = (𝑦 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘¦), (π‘”β€˜(2nd β€˜π‘¦))⟩)) ↔ (βˆƒπ‘“ ∈ ran (mRSubstβ€˜π‘‡)𝐹 = (π‘₯ ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘₯), (π‘“β€˜(2nd β€˜π‘₯))⟩) ∧ βˆƒπ‘” ∈ ran (mRSubstβ€˜π‘‡)𝐺 = (𝑦 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘¦), (π‘”β€˜(2nd β€˜π‘¦))⟩)))
18 simpr 483 . . . . . . . . . . . 12 (((𝑓 ∈ ran (mRSubstβ€˜π‘‡) ∧ 𝑔 ∈ ran (mRSubstβ€˜π‘‡)) ∧ 𝑦 ∈ (mExβ€˜π‘‡)) β†’ 𝑦 ∈ (mExβ€˜π‘‡))
19 eqid 2730 . . . . . . . . . . . . 13 (mTCβ€˜π‘‡) = (mTCβ€˜π‘‡)
20 eqid 2730 . . . . . . . . . . . . 13 (mRExβ€˜π‘‡) = (mRExβ€˜π‘‡)
2119, 1, 20mexval 34789 . . . . . . . . . . . 12 (mExβ€˜π‘‡) = ((mTCβ€˜π‘‡) Γ— (mRExβ€˜π‘‡))
2218, 21eleqtrdi 2841 . . . . . . . . . . 11 (((𝑓 ∈ ran (mRSubstβ€˜π‘‡) ∧ 𝑔 ∈ ran (mRSubstβ€˜π‘‡)) ∧ 𝑦 ∈ (mExβ€˜π‘‡)) β†’ 𝑦 ∈ ((mTCβ€˜π‘‡) Γ— (mRExβ€˜π‘‡)))
23 xp1st 8011 . . . . . . . . . . 11 (𝑦 ∈ ((mTCβ€˜π‘‡) Γ— (mRExβ€˜π‘‡)) β†’ (1st β€˜π‘¦) ∈ (mTCβ€˜π‘‡))
2422, 23syl 17 . . . . . . . . . 10 (((𝑓 ∈ ran (mRSubstβ€˜π‘‡) ∧ 𝑔 ∈ ran (mRSubstβ€˜π‘‡)) ∧ 𝑦 ∈ (mExβ€˜π‘‡)) β†’ (1st β€˜π‘¦) ∈ (mTCβ€˜π‘‡))
252, 20mrsubf 34804 . . . . . . . . . . . 12 (𝑔 ∈ ran (mRSubstβ€˜π‘‡) β†’ 𝑔:(mRExβ€˜π‘‡)⟢(mRExβ€˜π‘‡))
2625ad2antlr 723 . . . . . . . . . . 11 (((𝑓 ∈ ran (mRSubstβ€˜π‘‡) ∧ 𝑔 ∈ ran (mRSubstβ€˜π‘‡)) ∧ 𝑦 ∈ (mExβ€˜π‘‡)) β†’ 𝑔:(mRExβ€˜π‘‡)⟢(mRExβ€˜π‘‡))
27 xp2nd 8012 . . . . . . . . . . . 12 (𝑦 ∈ ((mTCβ€˜π‘‡) Γ— (mRExβ€˜π‘‡)) β†’ (2nd β€˜π‘¦) ∈ (mRExβ€˜π‘‡))
2822, 27syl 17 . . . . . . . . . . 11 (((𝑓 ∈ ran (mRSubstβ€˜π‘‡) ∧ 𝑔 ∈ ran (mRSubstβ€˜π‘‡)) ∧ 𝑦 ∈ (mExβ€˜π‘‡)) β†’ (2nd β€˜π‘¦) ∈ (mRExβ€˜π‘‡))
2926, 28ffvelcdmd 7088 . . . . . . . . . 10 (((𝑓 ∈ ran (mRSubstβ€˜π‘‡) ∧ 𝑔 ∈ ran (mRSubstβ€˜π‘‡)) ∧ 𝑦 ∈ (mExβ€˜π‘‡)) β†’ (π‘”β€˜(2nd β€˜π‘¦)) ∈ (mRExβ€˜π‘‡))
30 opelxpi 5714 . . . . . . . . . 10 (((1st β€˜π‘¦) ∈ (mTCβ€˜π‘‡) ∧ (π‘”β€˜(2nd β€˜π‘¦)) ∈ (mRExβ€˜π‘‡)) β†’ ⟨(1st β€˜π‘¦), (π‘”β€˜(2nd β€˜π‘¦))⟩ ∈ ((mTCβ€˜π‘‡) Γ— (mRExβ€˜π‘‡)))
3124, 29, 30syl2anc 582 . . . . . . . . 9 (((𝑓 ∈ ran (mRSubstβ€˜π‘‡) ∧ 𝑔 ∈ ran (mRSubstβ€˜π‘‡)) ∧ 𝑦 ∈ (mExβ€˜π‘‡)) β†’ ⟨(1st β€˜π‘¦), (π‘”β€˜(2nd β€˜π‘¦))⟩ ∈ ((mTCβ€˜π‘‡) Γ— (mRExβ€˜π‘‡)))
3231, 21eleqtrrdi 2842 . . . . . . . 8 (((𝑓 ∈ ran (mRSubstβ€˜π‘‡) ∧ 𝑔 ∈ ran (mRSubstβ€˜π‘‡)) ∧ 𝑦 ∈ (mExβ€˜π‘‡)) β†’ ⟨(1st β€˜π‘¦), (π‘”β€˜(2nd β€˜π‘¦))⟩ ∈ (mExβ€˜π‘‡))
33 eqidd 2731 . . . . . . . 8 ((𝑓 ∈ ran (mRSubstβ€˜π‘‡) ∧ 𝑔 ∈ ran (mRSubstβ€˜π‘‡)) β†’ (𝑦 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘¦), (π‘”β€˜(2nd β€˜π‘¦))⟩) = (𝑦 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘¦), (π‘”β€˜(2nd β€˜π‘¦))⟩))
34 eqidd 2731 . . . . . . . 8 ((𝑓 ∈ ran (mRSubstβ€˜π‘‡) ∧ 𝑔 ∈ ran (mRSubstβ€˜π‘‡)) β†’ (π‘₯ ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘₯), (π‘“β€˜(2nd β€˜π‘₯))⟩) = (π‘₯ ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘₯), (π‘“β€˜(2nd β€˜π‘₯))⟩))
35 fvex 6905 . . . . . . . . . 10 (1st β€˜π‘¦) ∈ V
36 fvex 6905 . . . . . . . . . 10 (π‘”β€˜(2nd β€˜π‘¦)) ∈ V
3735, 36op1std 7989 . . . . . . . . 9 (π‘₯ = ⟨(1st β€˜π‘¦), (π‘”β€˜(2nd β€˜π‘¦))⟩ β†’ (1st β€˜π‘₯) = (1st β€˜π‘¦))
3835, 36op2ndd 7990 . . . . . . . . . 10 (π‘₯ = ⟨(1st β€˜π‘¦), (π‘”β€˜(2nd β€˜π‘¦))⟩ β†’ (2nd β€˜π‘₯) = (π‘”β€˜(2nd β€˜π‘¦)))
3938fveq2d 6896 . . . . . . . . 9 (π‘₯ = ⟨(1st β€˜π‘¦), (π‘”β€˜(2nd β€˜π‘¦))⟩ β†’ (π‘“β€˜(2nd β€˜π‘₯)) = (π‘“β€˜(π‘”β€˜(2nd β€˜π‘¦))))
4037, 39opeq12d 4882 . . . . . . . 8 (π‘₯ = ⟨(1st β€˜π‘¦), (π‘”β€˜(2nd β€˜π‘¦))⟩ β†’ ⟨(1st β€˜π‘₯), (π‘“β€˜(2nd β€˜π‘₯))⟩ = ⟨(1st β€˜π‘¦), (π‘“β€˜(π‘”β€˜(2nd β€˜π‘¦)))⟩)
4132, 33, 34, 40fmptco 7130 . . . . . . 7 ((𝑓 ∈ ran (mRSubstβ€˜π‘‡) ∧ 𝑔 ∈ ran (mRSubstβ€˜π‘‡)) β†’ ((π‘₯ ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘₯), (π‘“β€˜(2nd β€˜π‘₯))⟩) ∘ (𝑦 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘¦), (π‘”β€˜(2nd β€˜π‘¦))⟩)) = (𝑦 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘¦), (π‘“β€˜(π‘”β€˜(2nd β€˜π‘¦)))⟩))
42 fvco3 6991 . . . . . . . . . 10 ((𝑔:(mRExβ€˜π‘‡)⟢(mRExβ€˜π‘‡) ∧ (2nd β€˜π‘¦) ∈ (mRExβ€˜π‘‡)) β†’ ((𝑓 ∘ 𝑔)β€˜(2nd β€˜π‘¦)) = (π‘“β€˜(π‘”β€˜(2nd β€˜π‘¦))))
4326, 28, 42syl2anc 582 . . . . . . . . 9 (((𝑓 ∈ ran (mRSubstβ€˜π‘‡) ∧ 𝑔 ∈ ran (mRSubstβ€˜π‘‡)) ∧ 𝑦 ∈ (mExβ€˜π‘‡)) β†’ ((𝑓 ∘ 𝑔)β€˜(2nd β€˜π‘¦)) = (π‘“β€˜(π‘”β€˜(2nd β€˜π‘¦))))
4443opeq2d 4881 . . . . . . . 8 (((𝑓 ∈ ran (mRSubstβ€˜π‘‡) ∧ 𝑔 ∈ ran (mRSubstβ€˜π‘‡)) ∧ 𝑦 ∈ (mExβ€˜π‘‡)) β†’ ⟨(1st β€˜π‘¦), ((𝑓 ∘ 𝑔)β€˜(2nd β€˜π‘¦))⟩ = ⟨(1st β€˜π‘¦), (π‘“β€˜(π‘”β€˜(2nd β€˜π‘¦)))⟩)
4544mpteq2dva 5249 . . . . . . 7 ((𝑓 ∈ ran (mRSubstβ€˜π‘‡) ∧ 𝑔 ∈ ran (mRSubstβ€˜π‘‡)) β†’ (𝑦 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘¦), ((𝑓 ∘ 𝑔)β€˜(2nd β€˜π‘¦))⟩) = (𝑦 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘¦), (π‘“β€˜(π‘”β€˜(2nd β€˜π‘¦)))⟩))
4641, 45eqtr4d 2773 . . . . . 6 ((𝑓 ∈ ran (mRSubstβ€˜π‘‡) ∧ 𝑔 ∈ ran (mRSubstβ€˜π‘‡)) β†’ ((π‘₯ ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘₯), (π‘“β€˜(2nd β€˜π‘₯))⟩) ∘ (𝑦 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘¦), (π‘”β€˜(2nd β€˜π‘¦))⟩)) = (𝑦 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘¦), ((𝑓 ∘ 𝑔)β€˜(2nd β€˜π‘¦))⟩))
472mrsubco 34808 . . . . . . . 8 ((𝑓 ∈ ran (mRSubstβ€˜π‘‡) ∧ 𝑔 ∈ ran (mRSubstβ€˜π‘‡)) β†’ (𝑓 ∘ 𝑔) ∈ ran (mRSubstβ€˜π‘‡))
487mptex 7228 . . . . . . . 8 (𝑦 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘¦), ((𝑓 ∘ 𝑔)β€˜(2nd β€˜π‘¦))⟩) ∈ V
49 eqid 2730 . . . . . . . . 9 (β„Ž ∈ ran (mRSubstβ€˜π‘‡) ↦ (𝑦 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘¦), (β„Žβ€˜(2nd β€˜π‘¦))⟩)) = (β„Ž ∈ ran (mRSubstβ€˜π‘‡) ↦ (𝑦 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘¦), (β„Žβ€˜(2nd β€˜π‘¦))⟩))
50 fveq1 6891 . . . . . . . . . . 11 (β„Ž = (𝑓 ∘ 𝑔) β†’ (β„Žβ€˜(2nd β€˜π‘¦)) = ((𝑓 ∘ 𝑔)β€˜(2nd β€˜π‘¦)))
5150opeq2d 4881 . . . . . . . . . 10 (β„Ž = (𝑓 ∘ 𝑔) β†’ ⟨(1st β€˜π‘¦), (β„Žβ€˜(2nd β€˜π‘¦))⟩ = ⟨(1st β€˜π‘¦), ((𝑓 ∘ 𝑔)β€˜(2nd β€˜π‘¦))⟩)
5251mpteq2dv 5251 . . . . . . . . 9 (β„Ž = (𝑓 ∘ 𝑔) β†’ (𝑦 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘¦), (β„Žβ€˜(2nd β€˜π‘¦))⟩) = (𝑦 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘¦), ((𝑓 ∘ 𝑔)β€˜(2nd β€˜π‘¦))⟩))
5349, 52elrnmpt1s 5957 . . . . . . . 8 (((𝑓 ∘ 𝑔) ∈ ran (mRSubstβ€˜π‘‡) ∧ (𝑦 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘¦), ((𝑓 ∘ 𝑔)β€˜(2nd β€˜π‘¦))⟩) ∈ V) β†’ (𝑦 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘¦), ((𝑓 ∘ 𝑔)β€˜(2nd β€˜π‘¦))⟩) ∈ ran (β„Ž ∈ ran (mRSubstβ€˜π‘‡) ↦ (𝑦 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘¦), (β„Žβ€˜(2nd β€˜π‘¦))⟩)))
5447, 48, 53sylancl 584 . . . . . . 7 ((𝑓 ∈ ran (mRSubstβ€˜π‘‡) ∧ 𝑔 ∈ ran (mRSubstβ€˜π‘‡)) β†’ (𝑦 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘¦), ((𝑓 ∘ 𝑔)β€˜(2nd β€˜π‘¦))⟩) ∈ ran (β„Ž ∈ ran (mRSubstβ€˜π‘‡) ↦ (𝑦 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘¦), (β„Žβ€˜(2nd β€˜π‘¦))⟩)))
551, 2, 3elmsubrn 34815 . . . . . . 7 ran 𝑆 = ran (β„Ž ∈ ran (mRSubstβ€˜π‘‡) ↦ (𝑦 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘¦), (β„Žβ€˜(2nd β€˜π‘¦))⟩))
5654, 55eleqtrrdi 2842 . . . . . 6 ((𝑓 ∈ ran (mRSubstβ€˜π‘‡) ∧ 𝑔 ∈ ran (mRSubstβ€˜π‘‡)) β†’ (𝑦 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘¦), ((𝑓 ∘ 𝑔)β€˜(2nd β€˜π‘¦))⟩) ∈ ran 𝑆)
5746, 56eqeltrd 2831 . . . . 5 ((𝑓 ∈ ran (mRSubstβ€˜π‘‡) ∧ 𝑔 ∈ ran (mRSubstβ€˜π‘‡)) β†’ ((π‘₯ ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘₯), (π‘“β€˜(2nd β€˜π‘₯))⟩) ∘ (𝑦 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘¦), (π‘”β€˜(2nd β€˜π‘¦))⟩)) ∈ ran 𝑆)
58 coeq1 5858 . . . . . . 7 (𝐹 = (π‘₯ ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘₯), (π‘“β€˜(2nd β€˜π‘₯))⟩) β†’ (𝐹 ∘ 𝐺) = ((π‘₯ ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘₯), (π‘“β€˜(2nd β€˜π‘₯))⟩) ∘ 𝐺))
59 coeq2 5859 . . . . . . 7 (𝐺 = (𝑦 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘¦), (π‘”β€˜(2nd β€˜π‘¦))⟩) β†’ ((π‘₯ ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘₯), (π‘“β€˜(2nd β€˜π‘₯))⟩) ∘ 𝐺) = ((π‘₯ ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘₯), (π‘“β€˜(2nd β€˜π‘₯))⟩) ∘ (𝑦 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘¦), (π‘”β€˜(2nd β€˜π‘¦))⟩)))
6058, 59sylan9eq 2790 . . . . . 6 ((𝐹 = (π‘₯ ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘₯), (π‘“β€˜(2nd β€˜π‘₯))⟩) ∧ 𝐺 = (𝑦 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘¦), (π‘”β€˜(2nd β€˜π‘¦))⟩)) β†’ (𝐹 ∘ 𝐺) = ((π‘₯ ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘₯), (π‘“β€˜(2nd β€˜π‘₯))⟩) ∘ (𝑦 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘¦), (π‘”β€˜(2nd β€˜π‘¦))⟩)))
6160eleq1d 2816 . . . . 5 ((𝐹 = (π‘₯ ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘₯), (π‘“β€˜(2nd β€˜π‘₯))⟩) ∧ 𝐺 = (𝑦 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘¦), (π‘”β€˜(2nd β€˜π‘¦))⟩)) β†’ ((𝐹 ∘ 𝐺) ∈ ran 𝑆 ↔ ((π‘₯ ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘₯), (π‘“β€˜(2nd β€˜π‘₯))⟩) ∘ (𝑦 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘¦), (π‘”β€˜(2nd β€˜π‘¦))⟩)) ∈ ran 𝑆))
6257, 61syl5ibrcom 246 . . . 4 ((𝑓 ∈ ran (mRSubstβ€˜π‘‡) ∧ 𝑔 ∈ ran (mRSubstβ€˜π‘‡)) β†’ ((𝐹 = (π‘₯ ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘₯), (π‘“β€˜(2nd β€˜π‘₯))⟩) ∧ 𝐺 = (𝑦 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘¦), (π‘”β€˜(2nd β€˜π‘¦))⟩)) β†’ (𝐹 ∘ 𝐺) ∈ ran 𝑆))
6362rexlimivv 3197 . . 3 (βˆƒπ‘“ ∈ ran (mRSubstβ€˜π‘‡)βˆƒπ‘” ∈ ran (mRSubstβ€˜π‘‡)(𝐹 = (π‘₯ ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘₯), (π‘“β€˜(2nd β€˜π‘₯))⟩) ∧ 𝐺 = (𝑦 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘¦), (π‘”β€˜(2nd β€˜π‘¦))⟩)) β†’ (𝐹 ∘ 𝐺) ∈ ran 𝑆)
6417, 63sylbir 234 . 2 ((βˆƒπ‘“ ∈ ran (mRSubstβ€˜π‘‡)𝐹 = (π‘₯ ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘₯), (π‘“β€˜(2nd β€˜π‘₯))⟩) ∧ βˆƒπ‘” ∈ ran (mRSubstβ€˜π‘‡)𝐺 = (𝑦 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘¦), (π‘”β€˜(2nd β€˜π‘¦))⟩)) β†’ (𝐹 ∘ 𝐺) ∈ ran 𝑆)
6510, 16, 64syl2anb 596 1 ((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) β†’ (𝐹 ∘ 𝐺) ∈ ran 𝑆)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1539   ∈ wcel 2104  βˆƒwrex 3068  Vcvv 3472  βŸ¨cop 4635   ↦ cmpt 5232   Γ— cxp 5675  ran crn 5678   ∘ ccom 5681  βŸΆwf 6540  β€˜cfv 6544  1st c1st 7977  2nd c2nd 7978  mTCcmtc 34751  mRExcmrex 34753  mExcmex 34754  mRSubstcmrsub 34757  mSubstcmsub 34758
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7729  ax-cnex 11170  ax-resscn 11171  ax-1cn 11172  ax-icn 11173  ax-addcl 11174  ax-addrcl 11175  ax-mulcl 11176  ax-mulrcl 11177  ax-mulcom 11178  ax-addass 11179  ax-mulass 11180  ax-distr 11181  ax-i2m1 11182  ax-1ne0 11183  ax-1rid 11184  ax-rnegex 11185  ax-rrecex 11186  ax-cnre 11187  ax-pre-lttri 11188  ax-pre-lttrn 11189  ax-pre-ltadd 11190  ax-pre-mulgt0 11191
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  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-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  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-riota 7369  df-ov 7416  df-oprab 7417  df-mpo 7418  df-om 7860  df-1st 7979  df-2nd 7980  df-frecs 8270  df-wrecs 8301  df-recs 8375  df-rdg 8414  df-1o 8470  df-er 8707  df-map 8826  df-pm 8827  df-en 8944  df-dom 8945  df-sdom 8946  df-fin 8947  df-card 9938  df-pnf 11256  df-mnf 11257  df-xr 11258  df-ltxr 11259  df-le 11260  df-sub 11452  df-neg 11453  df-nn 12219  df-2 12281  df-n0 12479  df-xnn0 12551  df-z 12565  df-uz 12829  df-fz 13491  df-fzo 13634  df-seq 13973  df-hash 14297  df-word 14471  df-lsw 14519  df-concat 14527  df-s1 14552  df-substr 14597  df-pfx 14627  df-struct 17086  df-sets 17103  df-slot 17121  df-ndx 17133  df-base 17151  df-ress 17180  df-plusg 17216  df-0g 17393  df-gsum 17394  df-mgm 18567  df-sgrp 18646  df-mnd 18662  df-mhm 18707  df-submnd 18708  df-frmd 18768  df-vrmd 18769  df-mrex 34773  df-mex 34774  df-mrsub 34777  df-msub 34778
This theorem is referenced by:  mclsppslem  34870
  Copyright terms: Public domain W3C validator