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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 2818 | . . 3 ⊢ (mRSubst‘𝑇) = (mRSubst‘𝑇) | |
6 | 1, 2, 3, 4, 5 | msubval 32669 | . 2 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) → ((𝑆‘𝐹)‘𝑋) = 〈(1st ‘𝑋), (((mRSubst‘𝑇)‘𝐹)‘(2nd ‘𝑋))〉) |
7 | fvex 6676 | . . 3 ⊢ (1st ‘𝑋) ∈ V | |
8 | fvex 6676 | . . 3 ⊢ (((mRSubst‘𝑇)‘𝐹)‘(2nd ‘𝑋)) ∈ V | |
9 | 7, 8 | op1std 7688 | . 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 1079 = wceq 1528 ∈ wcel 2105 ⊆ wss 3933 〈cop 4563 ⟶wf 6344 ‘cfv 6348 1st c1st 7676 2nd c2nd 7677 mVRcmvar 32605 mRExcmrex 32610 mExcmex 32611 mRSubstcmrsub 32614 mSubstcmsub 32615 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7678 df-pm 8398 df-msub 32635 |
This theorem is referenced by: (None) |
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