msubrsub - Mathbox for Mario Carneiro

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Theorem msubrsub 35472

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 35471	. 2 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) → ((𝑆‘𝐹)‘𝑋) = ⟨(1^st ‘𝑋), ((𝑂‘𝐹)‘(2^nd ‘𝑋))⟩)
7		fvex 6914	. . 3 ⊢ (1^st ‘𝑋) ∈ V
8		fvex 6914	. . 3 ⊢ ((𝑂‘𝐹)‘(2^nd ‘𝑋)) ∈ V
9	7, 8	op2ndd 8018	. 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 1085 = wceq 1535 ∈ wcel 2104 ⊆ wss 3963 ⟨cop 4636 ⟶wf 6554 ‘cfv 6558 1^st c1st 8005 2^nd c2nd 8006 mVRcmvar 35407 mRExcmrex 35412 mExcmex 35413 mRSubstcmrsub 35416 mSubstcmsub 35417

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5366 ax-pr 5430 ax-un 7747

This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2536 df-eu 2565 df-clab 2711 df-cleq 2725 df-clel 2812 df-nfc 2888 df-ne 2937 df-ral 3058 df-rex 3067 df-reu 3377 df-rab 3433 df-v 3479 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4915 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-iota 6510 df-fun 6560 df-fn 6561 df-f 6562 df-f1 6563 df-fo 6564 df-f1o 6565 df-fv 6566 df-ov 7428 df-oprab 7429 df-mpo 7430 df-2nd 8008 df-pm 8862 df-msub 35437

This theorem is referenced by: (None)

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