Theorem msubfval 35722

Description: A substitution applied to an expression. (Contributed by Mario Carneiro, 18-Jul-2016.)

Hypotheses

Ref	Expression
msubffval.v	⊢ 𝑉 = (mVR‘𝑇)
msubffval.r	⊢ 𝑅 = (mREx‘𝑇)
msubffval.s	⊢ 𝑆 = (mSubst‘𝑇)
msubffval.e	⊢ 𝐸 = (mEx‘𝑇)
msubffval.o	⊢ 𝑂 = (mRSubst‘𝑇)

Assertion

Ref	Expression
msubfval	⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉) → (𝑆‘𝐹) = (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), ((𝑂‘𝐹)‘(2nd ‘𝑒))⟩))

Distinct variable groups:   𝑒,𝐸   𝑒,𝑂   𝑅,𝑒   𝑇,𝑒   𝑒,𝑉   𝐴,𝑒   𝑒,𝐹

Allowed substitution hint:   𝑆(𝑒)

Proof of Theorem msubfval

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		msubffval.v	. . . . . 6 ⊢ 𝑉 = (mVR‘𝑇)
2		msubffval.r	. . . . . 6 ⊢ 𝑅 = (mREx‘𝑇)
3		msubffval.s	. . . . . 6 ⊢ 𝑆 = (mSubst‘𝑇)
4		msubffval.e	. . . . . 6 ⊢ 𝐸 = (mEx‘𝑇)
5		msubffval.o	. . . . . 6 ⊢ 𝑂 = (mRSubst‘𝑇)
6	1, 2, 3, 4, 5	msubffval 35721	. . . . 5 ⊢ (𝑇 ∈ V → 𝑆 = (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), ((𝑂‘𝑓)‘(2nd ‘𝑒))⟩)))
7	6	adantr 480	. . . 4 ⊢ ((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) → 𝑆 = (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), ((𝑂‘𝑓)‘(2nd ‘𝑒))⟩)))
8		simplr 769	. . . . . . . 8 ⊢ ((((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) ∧ 𝑓 = 𝐹) ∧ 𝑒 ∈ 𝐸) → 𝑓 = 𝐹)
9	8	fveq2d 6838	. . . . . . 7 ⊢ ((((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) ∧ 𝑓 = 𝐹) ∧ 𝑒 ∈ 𝐸) → (𝑂‘𝑓) = (𝑂‘𝐹))
10	9	fveq1d 6836	. . . . . 6 ⊢ ((((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) ∧ 𝑓 = 𝐹) ∧ 𝑒 ∈ 𝐸) → ((𝑂‘𝑓)‘(2nd ‘𝑒)) = ((𝑂‘𝐹)‘(2nd ‘𝑒)))
11	10	opeq2d 4824	. . . . 5 ⊢ ((((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) ∧ 𝑓 = 𝐹) ∧ 𝑒 ∈ 𝐸) → ⟨(1st ‘𝑒), ((𝑂‘𝑓)‘(2nd ‘𝑒))⟩ = ⟨(1st ‘𝑒), ((𝑂‘𝐹)‘(2nd ‘𝑒))⟩)
12	11	mpteq2dva 5179	. . . 4 ⊢ (((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) ∧ 𝑓 = 𝐹) → (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), ((𝑂‘𝑓)‘(2nd ‘𝑒))⟩) = (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), ((𝑂‘𝐹)‘(2nd ‘𝑒))⟩))
13	2	fvexi 6848	. . . . . . 7 ⊢ 𝑅 ∈ V
14	1	fvexi 6848	. . . . . . 7 ⊢ 𝑉 ∈ V
15	13, 14	pm3.2i 470	. . . . . 6 ⊢ (𝑅 ∈ V ∧ 𝑉 ∈ V)
16	15	a1i 11	. . . . 5 ⊢ (𝑇 ∈ V → (𝑅 ∈ V ∧ 𝑉 ∈ V))
17		elpm2r 8785	. . . . 5 ⊢ (((𝑅 ∈ V ∧ 𝑉 ∈ V) ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) → 𝐹 ∈ (𝑅 ↑pm 𝑉))
18	16, 17	sylan 581	. . . 4 ⊢ ((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) → 𝐹 ∈ (𝑅 ↑pm 𝑉))
19	4	fvexi 6848	. . . . . 6 ⊢ 𝐸 ∈ V
20	19	mptex 7171	. . . . 5 ⊢ (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), ((𝑂‘𝐹)‘(2nd ‘𝑒))⟩) ∈ V
21	20	a1i 11	. . . 4 ⊢ ((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) → (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), ((𝑂‘𝐹)‘(2nd ‘𝑒))⟩) ∈ V)
22	7, 12, 18, 21	fvmptd 6949	. . 3 ⊢ ((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) → (𝑆‘𝐹) = (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), ((𝑂‘𝐹)‘(2nd ‘𝑒))⟩))
23	22	ex 412	. 2 ⊢ (𝑇 ∈ V → ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉) → (𝑆‘𝐹) = (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), ((𝑂‘𝐹)‘(2nd ‘𝑒))⟩)))
24		0fv 6875	. . . . 5 ⊢ (∅‘𝐹) = ∅
25		mpt0 6634	. . . . 5 ⊢ (𝑒 ∈ ∅ ↦ ⟨(1st ‘𝑒), ((𝑂‘𝐹)‘(2nd ‘𝑒))⟩) = ∅
26	24, 25	eqtr4i 2763	. . . 4 ⊢ (∅‘𝐹) = (𝑒 ∈ ∅ ↦ ⟨(1st ‘𝑒), ((𝑂‘𝐹)‘(2nd ‘𝑒))⟩)
27		fvprc 6826	. . . . . 6 ⊢ (¬ 𝑇 ∈ V → (mSubst‘𝑇) = ∅)
28	3, 27	eqtrid 2784	. . . . 5 ⊢ (¬ 𝑇 ∈ V → 𝑆 = ∅)
29	28	fveq1d 6836	. . . 4 ⊢ (¬ 𝑇 ∈ V → (𝑆‘𝐹) = (∅‘𝐹))
30		fvprc 6826	. . . . . 6 ⊢ (¬ 𝑇 ∈ V → (mEx‘𝑇) = ∅)
31	4, 30	eqtrid 2784	. . . . 5 ⊢ (¬ 𝑇 ∈ V → 𝐸 = ∅)
32	31	mpteq1d 5176	. . . 4 ⊢ (¬ 𝑇 ∈ V → (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), ((𝑂‘𝐹)‘(2nd ‘𝑒))⟩) = (𝑒 ∈ ∅ ↦ ⟨(1st ‘𝑒), ((𝑂‘𝐹)‘(2nd ‘𝑒))⟩))
33	26, 29, 32	3eqtr4a 2798	. . 3 ⊢ (¬ 𝑇 ∈ V → (𝑆‘𝐹) = (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), ((𝑂‘𝐹)‘(2nd ‘𝑒))⟩))
34	33	a1d 25	. 2 ⊢ (¬ 𝑇 ∈ V → ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉) → (𝑆‘𝐹) = (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), ((𝑂‘𝐹)‘(2nd ‘𝑒))⟩)))
35	23, 34	pm2.61i 182	1 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉) → (𝑆‘𝐹) = (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), ((𝑂‘𝐹)‘(2nd ‘𝑒))⟩))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  Vcvv 3430   ⊆ wss 3890  ∅c0 4274  ⟨cop 4574   ↦ cmpt 5167  ⟶wf 6488  ‘cfv 6492  (class class class)co 7360  1^st c1st 7933  2^nd c2nd 7934   ↑_pm cpm 8767  mVRcmvar 35659  mRExcmrex 35664  mExcmex 35665  mRSubstcmrsub 35668  mSubstcmsub 35669

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pow 5302  ax-pr 5370  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-reu 3344  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-pw 4544  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  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 7363  df-oprab 7364  df-mpo 7365  df-pm 8769  df-msub 35689

This theorem is referenced by:  msubval  35723  msubrn  35727