msubfval - Mathbox for Mario Carneiro

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Theorem msubfval 34510

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 ↦ ⟨(1^st ‘𝑒), ((𝑂‘𝐹)‘(2^nd ‘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 34509	. . . . 5 ⊢ (𝑇 ∈ V → 𝑆 = (𝑓 ∈ (𝑅 ↑_pm 𝑉) ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), ((𝑂‘𝑓)‘(2^nd ‘𝑒))⟩)))
7	6	adantr 481	. . . 4 ⊢ ((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) → 𝑆 = (𝑓 ∈ (𝑅 ↑_pm 𝑉) ↦ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), ((𝑂‘𝑓)‘(2^nd ‘𝑒))⟩)))
8		simplr 767	. . . . . . . 8 ⊢ ((((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) ∧ 𝑓 = 𝐹) ∧ 𝑒 ∈ 𝐸) → 𝑓 = 𝐹)
9	8	fveq2d 6895	. . . . . . 7 ⊢ ((((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) ∧ 𝑓 = 𝐹) ∧ 𝑒 ∈ 𝐸) → (𝑂‘𝑓) = (𝑂‘𝐹))
10	9	fveq1d 6893	. . . . . 6 ⊢ ((((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) ∧ 𝑓 = 𝐹) ∧ 𝑒 ∈ 𝐸) → ((𝑂‘𝑓)‘(2^nd ‘𝑒)) = ((𝑂‘𝐹)‘(2^nd ‘𝑒)))
11	10	opeq2d 4880	. . . . 5 ⊢ ((((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) ∧ 𝑓 = 𝐹) ∧ 𝑒 ∈ 𝐸) → ⟨(1^st ‘𝑒), ((𝑂‘𝑓)‘(2^nd ‘𝑒))⟩ = ⟨(1^st ‘𝑒), ((𝑂‘𝐹)‘(2^nd ‘𝑒))⟩)
12	11	mpteq2dva 5248	. . . 4 ⊢ (((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) ∧ 𝑓 = 𝐹) → (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), ((𝑂‘𝑓)‘(2^nd ‘𝑒))⟩) = (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), ((𝑂‘𝐹)‘(2^nd ‘𝑒))⟩))
13	2	fvexi 6905	. . . . . . 7 ⊢ 𝑅 ∈ V
14	1	fvexi 6905	. . . . . . 7 ⊢ 𝑉 ∈ V
15	13, 14	pm3.2i 471	. . . . . 6 ⊢ (𝑅 ∈ V ∧ 𝑉 ∈ V)
16	15	a1i 11	. . . . 5 ⊢ (𝑇 ∈ V → (𝑅 ∈ V ∧ 𝑉 ∈ V))
17		elpm2r 8838	. . . . 5 ⊢ (((𝑅 ∈ V ∧ 𝑉 ∈ V) ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) → 𝐹 ∈ (𝑅 ↑_pm 𝑉))
18	16, 17	sylan 580	. . . 4 ⊢ ((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) → 𝐹 ∈ (𝑅 ↑_pm 𝑉))
19	4	fvexi 6905	. . . . . 6 ⊢ 𝐸 ∈ V
20	19	mptex 7224	. . . . 5 ⊢ (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), ((𝑂‘𝐹)‘(2^nd ‘𝑒))⟩) ∈ V
21	20	a1i 11	. . . 4 ⊢ ((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) → (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), ((𝑂‘𝐹)‘(2^nd ‘𝑒))⟩) ∈ V)
22	7, 12, 18, 21	fvmptd 7005	. . 3 ⊢ ((𝑇 ∈ V ∧ (𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉)) → (𝑆‘𝐹) = (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), ((𝑂‘𝐹)‘(2^nd ‘𝑒))⟩))
23	22	ex 413	. 2 ⊢ (𝑇 ∈ V → ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉) → (𝑆‘𝐹) = (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), ((𝑂‘𝐹)‘(2^nd ‘𝑒))⟩)))
24		0fv 6935	. . . . 5 ⊢ (∅‘𝐹) = ∅
25		mpt0 6692	. . . . 5 ⊢ (𝑒 ∈ ∅ ↦ ⟨(1^st ‘𝑒), ((𝑂‘𝐹)‘(2^nd ‘𝑒))⟩) = ∅
26	24, 25	eqtr4i 2763	. . . 4 ⊢ (∅‘𝐹) = (𝑒 ∈ ∅ ↦ ⟨(1^st ‘𝑒), ((𝑂‘𝐹)‘(2^nd ‘𝑒))⟩)
27		fvprc 6883	. . . . . 6 ⊢ (¬ 𝑇 ∈ V → (mSubst‘𝑇) = ∅)
28	3, 27	eqtrid 2784	. . . . 5 ⊢ (¬ 𝑇 ∈ V → 𝑆 = ∅)
29	28	fveq1d 6893	. . . 4 ⊢ (¬ 𝑇 ∈ V → (𝑆‘𝐹) = (∅‘𝐹))
30		fvprc 6883	. . . . . 6 ⊢ (¬ 𝑇 ∈ V → (mEx‘𝑇) = ∅)
31	4, 30	eqtrid 2784	. . . . 5 ⊢ (¬ 𝑇 ∈ V → 𝐸 = ∅)
32	31	mpteq1d 5243	. . . 4 ⊢ (¬ 𝑇 ∈ V → (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), ((𝑂‘𝐹)‘(2^nd ‘𝑒))⟩) = (𝑒 ∈ ∅ ↦ ⟨(1^st ‘𝑒), ((𝑂‘𝐹)‘(2^nd ‘𝑒))⟩))
33	26, 29, 32	3eqtr4a 2798	. . 3 ⊢ (¬ 𝑇 ∈ V → (𝑆‘𝐹) = (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), ((𝑂‘𝐹)‘(2^nd ‘𝑒))⟩))
34	33	a1d 25	. 2 ⊢ (¬ 𝑇 ∈ V → ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉) → (𝑆‘𝐹) = (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), ((𝑂‘𝐹)‘(2^nd ‘𝑒))⟩)))
35	23, 34	pm2.61i 182	1 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉) → (𝑆‘𝐹) = (𝑒 ∈ 𝐸 ↦ ⟨(1^st ‘𝑒), ((𝑂‘𝐹)‘(2^nd ‘𝑒))⟩))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3474 ⊆ wss 3948 ∅c0 4322 ⟨cop 4634 ↦ cmpt 5231 ⟶wf 6539 ‘cfv 6543 (class class class)co 7408 1^st c1st 7972 2^nd c2nd 7973 ↑_pm cpm 8820 mVRcmvar 34447 mRExcmrex 34452 mExcmex 34453 mRSubstcmrsub 34456 mSubstcmsub 34457

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 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7411 df-oprab 7412 df-mpo 7413 df-pm 8822 df-msub 34477

This theorem is referenced by: msubval 34511 msubrn 34515

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