msubty - Mathbox for Mario Carneiro

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Theorem msubty 35670

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 2734	. . 3 ⊢ (mRSubst‘𝑇) = (mRSubst‘𝑇)
6	1, 2, 3, 4, 5	msubval 35668	. 2 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) → ((𝑆‘𝐹)‘𝑋) = ⟨(1^st ‘𝑋), (((mRSubst‘𝑇)‘𝐹)‘(2^nd ‘𝑋))⟩)
7		fvex 6845	. . 3 ⊢ (1^st ‘𝑋) ∈ V
8		fvex 6845	. . 3 ⊢ (((mRSubst‘𝑇)‘𝐹)‘(2^nd ‘𝑋)) ∈ V
9	7, 8	op1std 7941	. 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 1086 = wceq 1541 ∈ wcel 2113 ⊆ wss 3899 ⟨cop 4584 ⟶wf 6486 ‘cfv 6490 1^st c1st 7929 2^nd c2nd 7930 mVRcmvar 35604 mRExcmrex 35609 mExcmex 35610 mRSubstcmrsub 35613 mSubstcmsub 35614

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 2115 ax-9 2123 ax-10 2146 ax-11 2162 ax-12 2182 ax-ext 2706 ax-rep 5222 ax-sep 5239 ax-nul 5249 ax-pow 5308 ax-pr 5375 ax-un 7678

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 2068 df-mo 2537 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2809 df-nfc 2883 df-ne 2931 df-ral 3050 df-rex 3059 df-reu 3349 df-rab 3398 df-v 3440 df-sbc 3739 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4579 df-pr 4581 df-op 4585 df-uni 4862 df-iun 4946 df-br 5097 df-opab 5159 df-mpt 5178 df-id 5517 df-xp 5628 df-rel 5629 df-cnv 5630 df-co 5631 df-dm 5632 df-rn 5633 df-res 5634 df-ima 5635 df-iota 6446 df-fun 6492 df-fn 6493 df-f 6494 df-f1 6495 df-fo 6496 df-f1o 6497 df-fv 6498 df-ov 7359 df-oprab 7360 df-mpo 7361 df-1st 7931 df-pm 8764 df-msub 35634

This theorem is referenced by: (None)

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