msubty - Mathbox for Mario Carneiro

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Theorem msubty 35073

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 2727	. . 3 ⊢ (mRSubst‘𝑇) = (mRSubst‘𝑇)
6	1, 2, 3, 4, 5	msubval 35071	. 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 7997	. 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 1085 = wceq 1534 ∈ wcel 2099 ⊆ wss 3944 ⟨cop 4630 ⟶wf 6538 ‘cfv 6542 1^st c1st 7985 2^nd c2nd 7986 mVRcmvar 35007 mRExcmrex 35012 mExcmex 35013 mRSubstcmrsub 35016 mSubstcmsub 35017

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 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-rep 5279 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-reu 3372 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-iun 4993 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7417 df-oprab 7418 df-mpo 7419 df-1st 7987 df-pm 8839 df-msub 35037

This theorem is referenced by: (None)

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