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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 35493 | . . 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 1131 | . 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 484 | . . . 4 ⊢ (((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝑅) ∧ 𝑒 = 𝑋) → 𝑒 = 𝑋) | |
9 | 8 | coeq2d 5876 | . . 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 7447 | . 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 1137 | . 2 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝑅) → 𝑋 ∈ 𝑅) | |
12 | ovexd 7466 | . 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 7023 | 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 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∪ cun 3961 ⊆ wss 3963 ifcif 4531 ↦ cmpt 5231 ∘ ccom 5693 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 〈“cs1 14630 Σg cgsu 17487 freeMndcfrmd 18873 mCNcmcn 35445 mVRcmvar 35446 mRExcmrex 35451 mRSubstcmrsub 35455 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-pm 8868 df-mrsub 35475 |
This theorem is referenced by: mrsubcv 35495 mrsub0 35501 mrsubccat 35503 |
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