msubty - Mathbox for Mario Carneiro

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Theorem msubty 34513

Description: The type of a substituted expression is the same as the original type. (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)

**Assertion**
Ref	Expression
msubty	⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) → (1^st ‘((𝑆‘𝐹)‘𝑋)) = (1^st ‘𝑋))

**Proof of Theorem msubty**
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		eqid 2732	. . 3 ⊢ (mRSubst‘𝑇) = (mRSubst‘𝑇)
6	1, 2, 3, 4, 5	msubval 34511	. 2 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) → ((𝑆‘𝐹)‘𝑋) = ⟨(1^st ‘𝑋), (((mRSubst‘𝑇)‘𝐹)‘(2^nd ‘𝑋))⟩)
7		fvex 6904	. . 3 ⊢ (1^st ‘𝑋) ∈ V
8		fvex 6904	. . 3 ⊢ (((mRSubst‘𝑇)‘𝐹)‘(2^nd ‘𝑋)) ∈ V
9	7, 8	op1std 7984	. 2 ⊢ (((𝑆‘𝐹)‘𝑋) = ⟨(1^st ‘𝑋), (((mRSubst‘𝑇)‘𝐹)‘(2^nd ‘𝑋))⟩ → (1^st ‘((𝑆‘𝐹)‘𝑋)) = (1^st ‘𝑋))
10	6, 9	syl 17	1 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) → (1^st ‘((𝑆‘𝐹)‘𝑋)) = (1^st ‘𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 ⊆ wss 3948 ⟨cop 4634 ⟶wf 6539 ‘cfv 6543 1^st c1st 7972 2^nd c2nd 7973 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-1st 7974 df-pm 8822 df-msub 34477

This theorem is referenced by: (None)

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