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Mirrors > Home > MPE Home > Th. List > Mathboxes > mrsubval | Structured version Visualization version GIF version |
Description: The substitution of some variables for expressions in a raw expression. (Contributed by Mario Carneiro, 18-Jul-2016.) |
Ref | Expression |
---|---|
mrsubffval.c | ⊢ 𝐶 = (mCN‘𝑇) |
mrsubffval.v | ⊢ 𝑉 = (mVR‘𝑇) |
mrsubffval.r | ⊢ 𝑅 = (mREx‘𝑇) |
mrsubffval.s | ⊢ 𝑆 = (mRSubst‘𝑇) |
mrsubffval.g | ⊢ 𝐺 = (freeMnd‘(𝐶 ∪ 𝑉)) |
Ref | Expression |
---|---|
mrsubval | ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝑅) → ((𝑆‘𝐹)‘𝑋) = (𝐺 Σg ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), 〈“𝑣”〉)) ∘ 𝑋))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mrsubffval.c | . . . 4 ⊢ 𝐶 = (mCN‘𝑇) | |
2 | mrsubffval.v | . . . 4 ⊢ 𝑉 = (mVR‘𝑇) | |
3 | mrsubffval.r | . . . 4 ⊢ 𝑅 = (mREx‘𝑇) | |
4 | mrsubffval.s | . . . 4 ⊢ 𝑆 = (mRSubst‘𝑇) | |
5 | mrsubffval.g | . . . 4 ⊢ 𝐺 = (freeMnd‘(𝐶 ∪ 𝑉)) | |
6 | 1, 2, 3, 4, 5 | mrsubfval 32868 | . . 3 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉) → (𝑆‘𝐹) = (𝑒 ∈ 𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), 〈“𝑣”〉)) ∘ 𝑒)))) |
7 | 6 | 3adant3 1129 | . 2 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝑅) → (𝑆‘𝐹) = (𝑒 ∈ 𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), 〈“𝑣”〉)) ∘ 𝑒)))) |
8 | simpr 488 | . . . 4 ⊢ (((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝑅) ∧ 𝑒 = 𝑋) → 𝑒 = 𝑋) | |
9 | 8 | coeq2d 5697 | . . 3 ⊢ (((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝑅) ∧ 𝑒 = 𝑋) → ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), 〈“𝑣”〉)) ∘ 𝑒) = ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), 〈“𝑣”〉)) ∘ 𝑋)) |
10 | 9 | oveq2d 7151 | . 2 ⊢ (((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝑅) ∧ 𝑒 = 𝑋) → (𝐺 Σg ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), 〈“𝑣”〉)) ∘ 𝑒)) = (𝐺 Σg ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), 〈“𝑣”〉)) ∘ 𝑋))) |
11 | simp3 1135 | . 2 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝑅) → 𝑋 ∈ 𝑅) | |
12 | ovexd 7170 | . 2 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝑅) → (𝐺 Σg ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), 〈“𝑣”〉)) ∘ 𝑋)) ∈ V) | |
13 | 7, 10, 11, 12 | fvmptd 6752 | 1 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝑅) → ((𝑆‘𝐹)‘𝑋) = (𝐺 Σg ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), 〈“𝑣”〉)) ∘ 𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∪ cun 3879 ⊆ wss 3881 ifcif 4425 ↦ cmpt 5110 ∘ ccom 5523 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 〈“cs1 13940 Σg cgsu 16706 freeMndcfrmd 18004 mCNcmcn 32820 mVRcmvar 32821 mRExcmrex 32826 mRSubstcmrsub 32830 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-pm 8392 df-mrsub 32850 |
This theorem is referenced by: mrsubcv 32870 mrsub0 32876 mrsubccat 32878 |
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