Theorem msubfval 31421

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 31420	. . . . 5 ⊢ (𝑇 ∈ V → 𝑆 = (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), ((𝑂‘𝑓)‘(2nd ‘𝑒))⟩)))
7	6	adantr 481	. . . 4 ⊢ ((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) → 𝑆 = (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), ((𝑂‘𝑓)‘(2nd ‘𝑒))⟩)))
8		simplr 792	. . . . . . . 8 ⊢ ((((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) ∧ 𝑓 = 𝐹) ∧ 𝑒 ∈ 𝐸) → 𝑓 = 𝐹)
9	8	fveq2d 6195	. . . . . . 7 ⊢ ((((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) ∧ 𝑓 = 𝐹) ∧ 𝑒 ∈ 𝐸) → (𝑂‘𝑓) = (𝑂‘𝐹))
10	9	fveq1d 6193	. . . . . 6 ⊢ ((((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) ∧ 𝑓 = 𝐹) ∧ 𝑒 ∈ 𝐸) → ((𝑂‘𝑓)‘(2nd ‘𝑒)) = ((𝑂‘𝐹)‘(2nd ‘𝑒)))
11	10	opeq2d 4409	. . . . 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 4744	. . . 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		fvex 6201	. . . . . . . 8 ⊢ (mREx‘𝑇) ∈ V
14	2, 13	eqeltri 2697	. . . . . . 7 ⊢ 𝑅 ∈ V
15		fvex 6201	. . . . . . . 8 ⊢ (mVR‘𝑇) ∈ V
16	1, 15	eqeltri 2697	. . . . . . 7 ⊢ 𝑉 ∈ V
17	14, 16	pm3.2i 471	. . . . . 6 ⊢ (𝑅 ∈ V ∧ 𝑉 ∈ V)
18	17	a1i 11	. . . . 5 ⊢ (𝑇 ∈ V → (𝑅 ∈ V ∧ 𝑉 ∈ V))
19		elpm2r 7875	. . . . 5 ⊢ (((𝑅 ∈ V ∧ 𝑉 ∈ V) ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) → 𝐹 ∈ (𝑅 ↑pm 𝑉))
20	18, 19	sylan 488	. . . 4 ⊢ ((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) → 𝐹 ∈ (𝑅 ↑pm 𝑉))
21		fvex 6201	. . . . . . 7 ⊢ (mEx‘𝑇) ∈ V
22	4, 21	eqeltri 2697	. . . . . 6 ⊢ 𝐸 ∈ V
23	22	mptex 6486	. . . . 5 ⊢ (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), ((𝑂‘𝐹)‘(2nd ‘𝑒))⟩) ∈ V
24	23	a1i 11	. . . 4 ⊢ ((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) → (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), ((𝑂‘𝐹)‘(2nd ‘𝑒))⟩) ∈ V)
25	7, 12, 20, 24	fvmptd 6288	. . 3 ⊢ ((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) → (𝑆‘𝐹) = (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), ((𝑂‘𝐹)‘(2nd ‘𝑒))⟩))
26	25	ex 450	. 2 ⊢ (𝑇 ∈ V → ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉) → (𝑆‘𝐹) = (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), ((𝑂‘𝐹)‘(2nd ‘𝑒))⟩)))
27		0fv 6227	. . . . 5 ⊢ (∅‘𝐹) = ∅
28		mpt0 6021	. . . . 5 ⊢ (𝑒 ∈ ∅ ↦ ⟨(1st ‘𝑒), ((𝑂‘𝐹)‘(2nd ‘𝑒))⟩) = ∅
29	27, 28	eqtr4i 2647	. . . 4 ⊢ (∅‘𝐹) = (𝑒 ∈ ∅ ↦ ⟨(1st ‘𝑒), ((𝑂‘𝐹)‘(2nd ‘𝑒))⟩)
30		fvprc 6185	. . . . . 6 ⊢ (¬ 𝑇 ∈ V → (mSubst‘𝑇) = ∅)
31	3, 30	syl5eq 2668	. . . . 5 ⊢ (¬ 𝑇 ∈ V → 𝑆 = ∅)
32	31	fveq1d 6193	. . . 4 ⊢ (¬ 𝑇 ∈ V → (𝑆‘𝐹) = (∅‘𝐹))
33		fvprc 6185	. . . . . 6 ⊢ (¬ 𝑇 ∈ V → (mEx‘𝑇) = ∅)
34	4, 33	syl5eq 2668	. . . . 5 ⊢ (¬ 𝑇 ∈ V → 𝐸 = ∅)
35	34	mpteq1d 4738	. . . 4 ⊢ (¬ 𝑇 ∈ V → (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), ((𝑂‘𝐹)‘(2nd ‘𝑒))⟩) = (𝑒 ∈ ∅ ↦ ⟨(1st ‘𝑒), ((𝑂‘𝐹)‘(2nd ‘𝑒))⟩))
36	29, 32, 35	3eqtr4a 2682	. . 3 ⊢ (¬ 𝑇 ∈ V → (𝑆‘𝐹) = (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), ((𝑂‘𝐹)‘(2nd ‘𝑒))⟩))
37	36	a1d 25	. 2 ⊢ (¬ 𝑇 ∈ V → ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉) → (𝑆‘𝐹) = (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), ((𝑂‘𝐹)‘(2nd ‘𝑒))⟩)))
38	26, 37	pm2.61i 176	1 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉) → (𝑆‘𝐹) = (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), ((𝑂‘𝐹)‘(2nd ‘𝑒))⟩))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 384   = wceq 1483   ∈ wcel 1990  Vcvv 3200   ⊆ wss 3574  ∅c0 3915  ⟨cop 4183   ↦ cmpt 4729  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  1^st c1st 7166  2^nd c2nd 7167   ↑_pm cpm 7858  mVRcmvar 31358  mRExcmrex 31363  mExcmex 31364  mRSubstcmrsub 31367  mSubstcmsub 31368

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-pm 7860  df-msub 31388

This theorem is referenced by:  msubval  31422  msubrn  31426