msubty - Mathbox for Mario Carneiro

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Theorem msubty 32995

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 2759	. . 3 ⊢ (mRSubst‘𝑇) = (mRSubst‘𝑇)
6	1, 2, 3, 4, 5	msubval 32993	. 2 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) → ((𝑆‘𝐹)‘𝑋) = ⟨(1^st ‘𝑋), (((mRSubst‘𝑇)‘𝐹)‘(2^nd ‘𝑋))⟩)
7		fvex 6669	. . 3 ⊢ (1^st ‘𝑋) ∈ V
8		fvex 6669	. . 3 ⊢ (((mRSubst‘𝑇)‘𝐹)‘(2^nd ‘𝑋)) ∈ V
9	7, 8	op1std 7701	. 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 1539 ∈ wcel 2112 ⊆ wss 3859 ⟨cop 4526 ⟶wf 6329 ‘cfv 6333 1^st c1st 7689 2^nd c2nd 7690 mVRcmvar 32929 mRExcmrex 32934 mExcmex 32935 mRSubstcmrsub 32938 mSubstcmsub 32939

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1912 ax-6 1971 ax-7 2016 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2159 ax-12 2176 ax-ext 2730 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5232 ax-pr 5296 ax-un 7457

This theorem depends on definitions: df-bi 210 df-an 401 df-or 846 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2071 df-mo 2558 df-eu 2589 df-clab 2737 df-cleq 2751 df-clel 2831 df-nfc 2902 df-ne 2953 df-ral 3076 df-rex 3077 df-reu 3078 df-rab 3080 df-v 3412 df-sbc 3698 df-csb 3807 df-dif 3862 df-un 3864 df-in 3866 df-ss 3876 df-nul 4227 df-if 4419 df-pw 4494 df-sn 4521 df-pr 4523 df-op 4527 df-uni 4797 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5428 df-xp 5528 df-rel 5529 df-cnv 5530 df-co 5531 df-dm 5532 df-rn 5533 df-res 5534 df-ima 5535 df-iota 6292 df-fun 6335 df-fn 6336 df-f 6337 df-f1 6338 df-fo 6339 df-f1o 6340 df-fv 6341 df-ov 7151 df-oprab 7152 df-mpo 7153 df-1st 7691 df-pm 8417 df-msub 32959

This theorem is referenced by: (None)

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