Theorem mrsubfval 35743

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
mrsubfval	⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉) → (𝑆‘𝐹) = (𝑒 ∈ 𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))))

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

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

Proof of Theorem mrsubfval

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		mrsubffval.c	. . . . . 6 ⊢ 𝐶 = (mCN‘𝑇)
2		mrsubffval.v	. . . . . 6 ⊢ 𝑉 = (mVR‘𝑇)
3		mrsubffval.r	. . . . . 6 ⊢ 𝑅 = (mREx‘𝑇)
4		mrsubffval.s	. . . . . 6 ⊢ 𝑆 = (mRSubst‘𝑇)
5		mrsubffval.g	. . . . . 6 ⊢ 𝐺 = (freeMnd‘(𝐶 ∪ 𝑉))
6	1, 2, 3, 4, 5	mrsubffval 35742	. . . . 5 ⊢ (𝑇 ∈ V → 𝑆 = (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)))))
7	6	adantr 481	. . . 4 ⊢ ((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) → 𝑆 = (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)))))
8		dmeq 5852	. . . . . . . . . . 11 ⊢ (𝑓 = 𝐹 → dom 𝑓 = dom 𝐹)
9		fdm 6671	. . . . . . . . . . . 12 ⊢ (𝐹:𝐴⟶𝑅 → dom 𝐹 = 𝐴)
10	9	ad2antrl 734	. . . . . . . . . . 11 ⊢ ((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) → dom 𝐹 = 𝐴)
11	8, 10	sylan9eqr 2797	. . . . . . . . . 10 ⊢ (((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) ∧ 𝑓 = 𝐹) → dom 𝑓 = 𝐴)
12	11	eleq2d 2826	. . . . . . . . 9 ⊢ (((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) ∧ 𝑓 = 𝐹) → (𝑣 ∈ dom 𝑓 ↔ 𝑣 ∈ 𝐴))
13		simpr 485	. . . . . . . . . 10 ⊢ (((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) ∧ 𝑓 = 𝐹) → 𝑓 = 𝐹)
14	13	fveq1d 6836	. . . . . . . . 9 ⊢ (((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) ∧ 𝑓 = 𝐹) → (𝑓‘𝑣) = (𝐹‘𝑣))
15	12, 14	ifbieq1d 4486	. . . . . . . 8 ⊢ (((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) ∧ 𝑓 = 𝐹) → if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩) = if(𝑣 ∈ 𝐴, (𝐹‘𝑣), ⟨“𝑣”⟩))
16	15	mpteq2dv 5173	. . . . . . 7 ⊢ (((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) ∧ 𝑓 = 𝐹) → (𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩)) = (𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), ⟨“𝑣”⟩)))
17	16	coeq1d 5810	. . . . . 6 ⊢ (((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) ∧ 𝑓 = 𝐹) → ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒) = ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))
18	17	oveq2d 7379	. . . . 5 ⊢ (((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) ∧ 𝑓 = 𝐹) → (𝐺 Σg ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)) = (𝐺 Σg ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)))
19	18	mpteq2dv 5173	. . . 4 ⊢ (((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) ∧ 𝑓 = 𝐹) → (𝑒 ∈ 𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ dom 𝑓, (𝑓‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))) = (𝑒 ∈ 𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))))
20	3	fvexi 6848	. . . . . 6 ⊢ 𝑅 ∈ V
21	20	a1i 11	. . . . 5 ⊢ ((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) → 𝑅 ∈ V)
22	2	fvexi 6848	. . . . . 6 ⊢ 𝑉 ∈ V
23	22	a1i 11	. . . . 5 ⊢ ((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) → 𝑉 ∈ V)
24		simprl 776	. . . . 5 ⊢ ((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) → 𝐹:𝐴⟶𝑅)
25		simprr 778	. . . . 5 ⊢ ((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) → 𝐴 ⊆ 𝑉)
26		elpm2r 8789	. . . . 5 ⊢ (((𝑅 ∈ V ∧ 𝑉 ∈ V) ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) → 𝐹 ∈ (𝑅 ↑pm 𝑉))
27	21, 23, 24, 25, 26	syl22anc 844	. . . 4 ⊢ ((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) → 𝐹 ∈ (𝑅 ↑pm 𝑉))
28	20	mptex 7174	. . . . 5 ⊢ (𝑒 ∈ 𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))) ∈ V
29	28	a1i 11	. . . 4 ⊢ ((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) → (𝑒 ∈ 𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))) ∈ V)
30	7, 19, 27, 29	fvmptd 6950	. . 3 ⊢ ((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) → (𝑆‘𝐹) = (𝑒 ∈ 𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))))
31	30	ex 413	. 2 ⊢ (𝑇 ∈ V → ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉) → (𝑆‘𝐹) = (𝑒 ∈ 𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)))))
32		0fv 6875	. . . 4 ⊢ (∅‘𝐹) = ∅
33		fvprc 6826	. . . . . 6 ⊢ (¬ 𝑇 ∈ V → (mRSubst‘𝑇) = ∅)
34	4, 33	eqtrid 2787	. . . . 5 ⊢ (¬ 𝑇 ∈ V → 𝑆 = ∅)
35	34	fveq1d 6836	. . . 4 ⊢ (¬ 𝑇 ∈ V → (𝑆‘𝐹) = (∅‘𝐹))
36		fvprc 6826	. . . . . . 7 ⊢ (¬ 𝑇 ∈ V → (mREx‘𝑇) = ∅)
37	3, 36	eqtrid 2787	. . . . . 6 ⊢ (¬ 𝑇 ∈ V → 𝑅 = ∅)
38	37	mpteq1d 5169	. . . . 5 ⊢ (¬ 𝑇 ∈ V → (𝑒 ∈ 𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))) = (𝑒 ∈ ∅ ↦ (𝐺 Σg ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))))
39		mpt0 6634	. . . . 5 ⊢ (𝑒 ∈ ∅ ↦ (𝐺 Σg ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))) = ∅
40	38, 39	eqtrdi 2791	. . . 4 ⊢ (¬ 𝑇 ∈ V → (𝑒 ∈ 𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))) = ∅)
41	32, 35, 40	3eqtr4a 2801	. . 3 ⊢ (¬ 𝑇 ∈ V → (𝑆‘𝐹) = (𝑒 ∈ 𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))))
42	41	a1d 25	. 2 ⊢ (¬ 𝑇 ∈ V → ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉) → (𝑆‘𝐹) = (𝑒 ∈ 𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒)))))
43	31, 42	pm2.61i 183	1 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉) → (𝑆‘𝐹) = (𝑒 ∈ 𝑅 ↦ (𝐺 Σg ((𝑣 ∈ (𝐶 ∪ 𝑉) ↦ if(𝑣 ∈ 𝐴, (𝐹‘𝑣), ⟨“𝑣”⟩)) ∘ 𝑒))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   = wceq 1547   ∈ wcel 2119  Vcvv 3432   ∪ cun 3888   ⊆ wss 3890  ∅c0 4268  ifcif 4461   ↦ cmpt 5160  dom cdm 5625   ∘ ccom 5629  ⟶wf 6488  ‘cfv 6492  (class class class)co 7363   ↑_pm cpm 8771  ⟨“cs1 14556   Σ_g cgsu 17401  freeMndcfrmd 18813  mCNcmcn 35695  mVRcmvar 35696  mRExcmrex 35701  mRSubstcmrsub 35705

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-rep 5206  ax-sep 5225  ax-nul 5235  ax-pow 5301  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-reu 3346  df-rab 3393  df-v 3434  df-sbc 3731  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-pw 4538  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-iun 4930  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  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 7366  df-oprab 7367  df-mpo 7368  df-pm 8773  df-mrsub 35725

This theorem is referenced by:  mrsubval  35744  mrsubrn  35748  elmrsubrn  35755