Theorem msubfval 30675

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 30674	. . . . 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 788	. . . . . . . 8 ⊢ ((((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) ∧ 𝑓 = 𝐹) ∧ 𝑒 ∈ 𝐸) → 𝑓 = 𝐹)
9	8	fveq2d 6107	. . . . . . 7 ⊢ ((((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) ∧ 𝑓 = 𝐹) ∧ 𝑒 ∈ 𝐸) → (𝑂‘𝑓) = (𝑂‘𝐹))
10	9	fveq1d 6105	. . . . . 6 ⊢ ((((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) ∧ 𝑓 = 𝐹) ∧ 𝑒 ∈ 𝐸) → ((𝑂‘𝑓)‘(2nd ‘𝑒)) = ((𝑂‘𝐹)‘(2nd ‘𝑒)))
11	10	opeq2d 4347	. . . . 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 4672	. . . 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 6113	. . . . . . . 8 ⊢ (mREx‘𝑇) ∈ V
14	2, 13	eqeltri 2684	. . . . . . 7 ⊢ 𝑅 ∈ V
15		fvex 6113	. . . . . . . 8 ⊢ (mVR‘𝑇) ∈ V
16	1, 15	eqeltri 2684	. . . . . . 7 ⊢ 𝑉 ∈ V
17	14, 16	pm3.2i 470	. . . . . 6 ⊢ (𝑅 ∈ V ∧ 𝑉 ∈ V)
18	17	a1i 11	. . . . 5 ⊢ (𝑇 ∈ V → (𝑅 ∈ V ∧ 𝑉 ∈ V))
19		elpm2r 7761	. . . . 5 ⊢ (((𝑅 ∈ V ∧ 𝑉 ∈ V) ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) → 𝐹 ∈ (𝑅 ↑pm 𝑉))
20	18, 19	sylan 487	. . . 4 ⊢ ((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) → 𝐹 ∈ (𝑅 ↑pm 𝑉))
21		fvex 6113	. . . . . . 7 ⊢ (mEx‘𝑇) ∈ V
22	4, 21	eqeltri 2684	. . . . . 6 ⊢ 𝐸 ∈ V
23	22	mptex 6390	. . . . 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 6197	. . 3 ⊢ ((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) → (𝑆‘𝐹) = (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), ((𝑂‘𝐹)‘(2nd ‘𝑒))⟩))
26	25	ex 449	. 2 ⊢ (𝑇 ∈ V → ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉) → (𝑆‘𝐹) = (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), ((𝑂‘𝐹)‘(2nd ‘𝑒))⟩)))
27		0fv 6137	. . . . 5 ⊢ (∅‘𝐹) = ∅
28		mpt0 5934	. . . . 5 ⊢ (𝑒 ∈ ∅ ↦ ⟨(1st ‘𝑒), ((𝑂‘𝐹)‘(2nd ‘𝑒))⟩) = ∅
29	27, 28	eqtr4i 2635	. . . 4 ⊢ (∅‘𝐹) = (𝑒 ∈ ∅ ↦ ⟨(1st ‘𝑒), ((𝑂‘𝐹)‘(2nd ‘𝑒))⟩)
30		fvprc 6097	. . . . . 6 ⊢ (¬ 𝑇 ∈ V → (mSubst‘𝑇) = ∅)
31	3, 30	syl5eq 2656	. . . . 5 ⊢ (¬ 𝑇 ∈ V → 𝑆 = ∅)
32	31	fveq1d 6105	. . . 4 ⊢ (¬ 𝑇 ∈ V → (𝑆‘𝐹) = (∅‘𝐹))
33		fvprc 6097	. . . . . 6 ⊢ (¬ 𝑇 ∈ V → (mEx‘𝑇) = ∅)
34	4, 33	syl5eq 2656	. . . . 5 ⊢ (¬ 𝑇 ∈ V → 𝐸 = ∅)
35	34	mpteq1d 4666	. . . 4 ⊢ (¬ 𝑇 ∈ V → (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), ((𝑂‘𝐹)‘(2nd ‘𝑒))⟩) = (𝑒 ∈ ∅ ↦ ⟨(1st ‘𝑒), ((𝑂‘𝐹)‘(2nd ‘𝑒))⟩))
36	29, 32, 35	3eqtr4a 2670	. . 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 175	1 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉) → (𝑆‘𝐹) = (𝑒 ∈ 𝐸 ↦ ⟨(1st ‘𝑒), ((𝑂‘𝐹)‘(2nd ‘𝑒))⟩))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540  ∅c0 3874  ⟨cop 4131   ↦ cmpt 4643  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  1^st c1st 7057  2^nd c2nd 7058   ↑_pm cpm 7745  mVRcmvar 30612  mRExcmrex 30617  mExcmex 30618  mRSubstcmrsub 30621  mSubstcmsub 30622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-pm 7747  df-msub 30642

This theorem is referenced by:  msubval  30676  msubrn  30680