msubrsub - Mathbox for Mario Carneiro

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Theorem msubrsub 35538

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
msubrsub	⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) → (2^nd ‘((𝑆‘𝐹)‘𝑋)) = ((𝑂‘𝐹)‘(2^nd ‘𝑋)))

**Proof of Theorem msubrsub**
Step	Hyp	Ref	Expression
1		msubffval.v	. . 3 ⊢ 𝑉 = (mVR‘𝑇)
2		msubffval.r	. . 3 ⊢ 𝑅 = (mREx‘𝑇)
3		msubffval.s	. . 3 ⊢ 𝑆 = (mSubst‘𝑇)
4		msubffval.e	. . 3 ⊢ 𝐸 = (mEx‘𝑇)
5		msubffval.o	. . 3 ⊢ 𝑂 = (mRSubst‘𝑇)
6	1, 2, 3, 4, 5	msubval 35537	. 2 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) → ((𝑆‘𝐹)‘𝑋) = ⟨(1^st ‘𝑋), ((𝑂‘𝐹)‘(2^nd ‘𝑋))⟩)
7		fvex 6830	. . 3 ⊢ (1^st ‘𝑋) ∈ V
8		fvex 6830	. . 3 ⊢ ((𝑂‘𝐹)‘(2^nd ‘𝑋)) ∈ V
9	7, 8	op2ndd 7927	. 2 ⊢ (((𝑆‘𝐹)‘𝑋) = ⟨(1^st ‘𝑋), ((𝑂‘𝐹)‘(2^nd ‘𝑋))⟩ → (2^nd ‘((𝑆‘𝐹)‘𝑋)) = ((𝑂‘𝐹)‘(2^nd ‘𝑋)))
10	6, 9	syl 17	1 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) → (2^nd ‘((𝑆‘𝐹)‘𝑋)) = ((𝑂‘𝐹)‘(2^nd ‘𝑋)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1086 = wceq 1541 ∈ wcel 2110 ⊆ wss 3900 ⟨cop 4580 ⟶wf 6473 ‘cfv 6477 1^st c1st 7914 2^nd c2nd 7915 mVRcmvar 35473 mRExcmrex 35478 mExcmex 35479 mRSubstcmrsub 35482 mSubstcmsub 35483

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1968 ax-7 2009 ax-8 2112 ax-9 2120 ax-10 2143 ax-11 2159 ax-12 2179 ax-ext 2702 ax-rep 5215 ax-sep 5232 ax-nul 5242 ax-pow 5301 ax-pr 5368 ax-un 7663

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1544 df-fal 1554 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-reu 3345 df-rab 3394 df-v 3436 df-sbc 3740 df-csb 3849 df-dif 3903 df-un 3905 df-in 3907 df-ss 3917 df-nul 4282 df-if 4474 df-pw 4550 df-sn 4575 df-pr 4577 df-op 4581 df-uni 4858 df-iun 4941 df-br 5090 df-opab 5152 df-mpt 5171 df-id 5509 df-xp 5620 df-rel 5621 df-cnv 5622 df-co 5623 df-dm 5624 df-rn 5625 df-res 5626 df-ima 5627 df-iota 6433 df-fun 6479 df-fn 6480 df-f 6481 df-f1 6482 df-fo 6483 df-f1o 6484 df-fv 6485 df-ov 7344 df-oprab 7345 df-mpo 7346 df-2nd 7917 df-pm 8748 df-msub 35503

This theorem is referenced by: (None)

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