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Theorem msubco 34517
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 2732 . . . . 5 (mExβ€˜π‘‡) = (mExβ€˜π‘‡)
2 eqid 2732 . . . . 5 (mRSubstβ€˜π‘‡) = (mRSubstβ€˜π‘‡)
3 msubco.s . . . . 5 𝑆 = (mSubstβ€˜π‘‡)
41, 2, 3elmsubrn 34514 . . . 4 ran 𝑆 = ran (𝑓 ∈ ran (mRSubstβ€˜π‘‡) ↦ (π‘₯ ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘₯), (π‘“β€˜(2nd β€˜π‘₯))⟩))
54eleq2i 2825 . . 3 (𝐹 ∈ ran 𝑆 ↔ 𝐹 ∈ ran (𝑓 ∈ ran (mRSubstβ€˜π‘‡) ↦ (π‘₯ ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘₯), (π‘“β€˜(2nd β€˜π‘₯))⟩)))
6 eqid 2732 . . . 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 6904 . . . . 5 (mExβ€˜π‘‡) ∈ V
87mptex 7224 . . . 4 (π‘₯ ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘₯), (π‘“β€˜(2nd β€˜π‘₯))⟩) ∈ V
96, 8elrnmpti 5959 . . 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 34514 . . . 4 ran 𝑆 = ran (𝑔 ∈ ran (mRSubstβ€˜π‘‡) ↦ (𝑦 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘¦), (π‘”β€˜(2nd β€˜π‘¦))⟩))
1211eleq2i 2825 . . 3 (𝐺 ∈ ran 𝑆 ↔ 𝐺 ∈ ran (𝑔 ∈ ran (mRSubstβ€˜π‘‡) ↦ (𝑦 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘¦), (π‘”β€˜(2nd β€˜π‘¦))⟩)))
13 eqid 2732 . . . 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 7224 . . . 4 (𝑦 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘¦), (π‘”β€˜(2nd β€˜π‘¦))⟩) ∈ V
1513, 14elrnmpti 5959 . . 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 3226 . . 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 485 . . . . . . . . . . . 12 (((𝑓 ∈ ran (mRSubstβ€˜π‘‡) ∧ 𝑔 ∈ ran (mRSubstβ€˜π‘‡)) ∧ 𝑦 ∈ (mExβ€˜π‘‡)) β†’ 𝑦 ∈ (mExβ€˜π‘‡))
19 eqid 2732 . . . . . . . . . . . . 13 (mTCβ€˜π‘‡) = (mTCβ€˜π‘‡)
20 eqid 2732 . . . . . . . . . . . . 13 (mRExβ€˜π‘‡) = (mRExβ€˜π‘‡)
2119, 1, 20mexval 34488 . . . . . . . . . . . 12 (mExβ€˜π‘‡) = ((mTCβ€˜π‘‡) Γ— (mRExβ€˜π‘‡))
2218, 21eleqtrdi 2843 . . . . . . . . . . 11 (((𝑓 ∈ ran (mRSubstβ€˜π‘‡) ∧ 𝑔 ∈ ran (mRSubstβ€˜π‘‡)) ∧ 𝑦 ∈ (mExβ€˜π‘‡)) β†’ 𝑦 ∈ ((mTCβ€˜π‘‡) Γ— (mRExβ€˜π‘‡)))
23 xp1st 8006 . . . . . . . . . . 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 34503 . . . . . . . . . . . 12 (𝑔 ∈ ran (mRSubstβ€˜π‘‡) β†’ 𝑔:(mRExβ€˜π‘‡)⟢(mRExβ€˜π‘‡))
2625ad2antlr 725 . . . . . . . . . . 11 (((𝑓 ∈ ran (mRSubstβ€˜π‘‡) ∧ 𝑔 ∈ ran (mRSubstβ€˜π‘‡)) ∧ 𝑦 ∈ (mExβ€˜π‘‡)) β†’ 𝑔:(mRExβ€˜π‘‡)⟢(mRExβ€˜π‘‡))
27 xp2nd 8007 . . . . . . . . . . . 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 7087 . . . . . . . . . 10 (((𝑓 ∈ ran (mRSubstβ€˜π‘‡) ∧ 𝑔 ∈ ran (mRSubstβ€˜π‘‡)) ∧ 𝑦 ∈ (mExβ€˜π‘‡)) β†’ (π‘”β€˜(2nd β€˜π‘¦)) ∈ (mRExβ€˜π‘‡))
30 opelxpi 5713 . . . . . . . . . 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 584 . . . . . . . . 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 2844 . . . . . . . 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 2733 . . . . . . . 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 2733 . . . . . . . 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 6904 . . . . . . . . . 10 (1st β€˜π‘¦) ∈ V
36 fvex 6904 . . . . . . . . . 10 (π‘”β€˜(2nd β€˜π‘¦)) ∈ V
3735, 36op1std 7984 . . . . . . . . 9 (π‘₯ = ⟨(1st β€˜π‘¦), (π‘”β€˜(2nd β€˜π‘¦))⟩ β†’ (1st β€˜π‘₯) = (1st β€˜π‘¦))
3835, 36op2ndd 7985 . . . . . . . . . 10 (π‘₯ = ⟨(1st β€˜π‘¦), (π‘”β€˜(2nd β€˜π‘¦))⟩ β†’ (2nd β€˜π‘₯) = (π‘”β€˜(2nd β€˜π‘¦)))
3938fveq2d 6895 . . . . . . . . 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 4881 . . . . . . . 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 7126 . . . . . . 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 6990 . . . . . . . . . 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 584 . . . . . . . . 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 4880 . . . . . . . 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 5248 . . . . . . 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 2775 . . . . . 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 34507 . . . . . . . 8 ((𝑓 ∈ ran (mRSubstβ€˜π‘‡) ∧ 𝑔 ∈ ran (mRSubstβ€˜π‘‡)) β†’ (𝑓 ∘ 𝑔) ∈ ran (mRSubstβ€˜π‘‡))
487mptex 7224 . . . . . . . 8 (𝑦 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘¦), ((𝑓 ∘ 𝑔)β€˜(2nd β€˜π‘¦))⟩) ∈ V
49 eqid 2732 . . . . . . . . 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 6890 . . . . . . . . . . 11 (β„Ž = (𝑓 ∘ 𝑔) β†’ (β„Žβ€˜(2nd β€˜π‘¦)) = ((𝑓 ∘ 𝑔)β€˜(2nd β€˜π‘¦)))
5150opeq2d 4880 . . . . . . . . . 10 (β„Ž = (𝑓 ∘ 𝑔) β†’ ⟨(1st β€˜π‘¦), (β„Žβ€˜(2nd β€˜π‘¦))⟩ = ⟨(1st β€˜π‘¦), ((𝑓 ∘ 𝑔)β€˜(2nd β€˜π‘¦))⟩)
5251mpteq2dv 5250 . . . . . . . . 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 5956 . . . . . . . 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 586 . . . . . . 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 34514 . . . . . . 7 ran 𝑆 = ran (β„Ž ∈ ran (mRSubstβ€˜π‘‡) ↦ (𝑦 ∈ (mExβ€˜π‘‡) ↦ ⟨(1st β€˜π‘¦), (β„Žβ€˜(2nd β€˜π‘¦))⟩))
5654, 55eleqtrrdi 2844 . . . . . 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 2833 . . . . 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 5857 . . . . . . 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 5858 . . . . . . 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 2792 . . . . . 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 2818 . . . . 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 3199 . . 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 598 1 ((𝐹 ∈ ran 𝑆 ∧ 𝐺 ∈ ran 𝑆) β†’ (𝐹 ∘ 𝐺) ∈ ran 𝑆)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆƒwrex 3070  Vcvv 3474  βŸ¨cop 4634   ↦ cmpt 5231   Γ— cxp 5674  ran crn 5677   ∘ ccom 5680  βŸΆwf 6539  β€˜cfv 6543  1st c1st 7972  2nd c2nd 7973  mTCcmtc 34450  mRExcmrex 34452  mExcmex 34453  mRSubstcmrsub 34456  mSubstcmsub 34457
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  ax-cnex 11165  ax-resscn 11166  ax-1cn 11167  ax-icn 11168  ax-addcl 11169  ax-addrcl 11170  ax-mulcl 11171  ax-mulrcl 11172  ax-mulcom 11173  ax-addass 11174  ax-mulass 11175  ax-distr 11176  ax-i2m1 11177  ax-1ne0 11178  ax-1rid 11179  ax-rnegex 11180  ax-rrecex 11181  ax-cnre 11182  ax-pre-lttri 11183  ax-pre-lttrn 11184  ax-pre-ltadd 11185  ax-pre-mulgt0 11186
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  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-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3376  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-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  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-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  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-riota 7364  df-ov 7411  df-oprab 7412  df-mpo 7413  df-om 7855  df-1st 7974  df-2nd 7975  df-frecs 8265  df-wrecs 8296  df-recs 8370  df-rdg 8409  df-1o 8465  df-er 8702  df-map 8821  df-pm 8822  df-en 8939  df-dom 8940  df-sdom 8941  df-fin 8942  df-card 9933  df-pnf 11249  df-mnf 11250  df-xr 11251  df-ltxr 11252  df-le 11253  df-sub 11445  df-neg 11446  df-nn 12212  df-2 12274  df-n0 12472  df-xnn0 12544  df-z 12558  df-uz 12822  df-fz 13484  df-fzo 13627  df-seq 13966  df-hash 14290  df-word 14464  df-lsw 14512  df-concat 14520  df-s1 14545  df-substr 14590  df-pfx 14620  df-struct 17079  df-sets 17096  df-slot 17114  df-ndx 17126  df-base 17144  df-ress 17173  df-plusg 17209  df-0g 17386  df-gsum 17387  df-mgm 18560  df-sgrp 18609  df-mnd 18625  df-mhm 18670  df-submnd 18671  df-frmd 18729  df-vrmd 18730  df-mrex 34472  df-mex 34473  df-mrsub 34476  df-msub 34477
This theorem is referenced by:  mclsppslem  34569
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