Theorem mrsubval 35703

Description: The substitution of some variables for expressions in a raw expression. (Contributed by Mario Carneiro, 18-Jul-2016.)

Hypotheses

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))

Assertion

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)))

Distinct variable groups:   𝑣,𝐴   𝑣,𝐶   𝑣,𝐹   𝑣,𝑅   𝑣,𝑋   𝑣,𝑇   𝑣,𝑉

Allowed substitution hints:   𝑆(𝑣)   𝐺(𝑣)

Proof of Theorem mrsubval

Dummy variable 𝑒 is distinct from all other variables.

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 35702	. . 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 1132	. 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 5811	. . 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 7374	. 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 1138	. 2 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝑅) → 𝑋 ∈ 𝑅)
12		ovexd 7393	. 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 6948	1 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝑅) → ((𝑆‘𝐹)‘𝑋) = (𝐺 Σg ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑋)))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2113  Vcvv 3440   ∪ cun 3899   ⊆ wss 3901  ifcif 4479   ↦ cmpt 5179   ∘ ccom 5628  ⟶wf 6488  ‘cfv 6492  (class class class)co 7358  ⟨“cs1 14519   Σ_g cgsu 17360  freeMndcfrmd 18772  mCNcmcn 35654  mVRcmvar 35655  mRExcmrex 35660  mRSubstcmrsub 35664

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-ov 7361  df-oprab 7362  df-mpo 7363  df-pm 8766  df-mrsub 35684

This theorem is referenced by:  mrsubcv  35704  mrsub0  35710  mrsubccat  35712