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Mathbox for Mario Carneiro |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > msubty | Structured version Visualization version GIF version |
Description: The type of a substituted expression is the same as the original type. (Contributed by Mario Carneiro, 18-Jul-2016.) |
Ref | Expression |
---|---|
msubffval.v | ⊢ 𝑉 = (mVR‘𝑇) |
msubffval.r | ⊢ 𝑅 = (mREx‘𝑇) |
msubffval.s | ⊢ 𝑆 = (mSubst‘𝑇) |
msubffval.e | ⊢ 𝐸 = (mEx‘𝑇) |
Ref | Expression |
---|---|
msubty | ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) → (1st ‘((𝑆‘𝐹)‘𝑋)) = (1st ‘𝑋)) |
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 2798 | . . 3 ⊢ (mRSubst‘𝑇) = (mRSubst‘𝑇) | |
6 | 1, 2, 3, 4, 5 | msubval 32885 | . 2 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) → ((𝑆‘𝐹)‘𝑋) = 〈(1st ‘𝑋), (((mRSubst‘𝑇)‘𝐹)‘(2nd ‘𝑋))〉) |
7 | fvex 6658 | . . 3 ⊢ (1st ‘𝑋) ∈ V | |
8 | fvex 6658 | . . 3 ⊢ (((mRSubst‘𝑇)‘𝐹)‘(2nd ‘𝑋)) ∈ V | |
9 | 7, 8 | op1std 7681 | . 2 ⊢ (((𝑆‘𝐹)‘𝑋) = 〈(1st ‘𝑋), (((mRSubst‘𝑇)‘𝐹)‘(2nd ‘𝑋))〉 → (1st ‘((𝑆‘𝐹)‘𝑋)) = (1st ‘𝑋)) |
10 | 6, 9 | syl 17 | 1 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) → (1st ‘((𝑆‘𝐹)‘𝑋)) = (1st ‘𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ⊆ wss 3881 〈cop 4531 ⟶wf 6320 ‘cfv 6324 1st c1st 7669 2nd c2nd 7670 mVRcmvar 32821 mRExcmrex 32826 mExcmex 32827 mRSubstcmrsub 32830 mSubstcmsub 32831 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-pm 8392 df-msub 32851 |
This theorem is referenced by: (None) |
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