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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 2756 | . . 3 ⊢ (mRSubst‘𝑇) = (mRSubst‘𝑇) | |
| 6 | 1, 2, 3, 4, 5 | msubval 35823 | . 2 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) → ((𝑆‘𝐹)‘𝑋) = 〈(1st ‘𝑋), (((mRSubst‘𝑇)‘𝐹)‘(2nd ‘𝑋))〉) |
| 7 | fvex 6869 | . . 3 ⊢ (1st ‘𝑋) ∈ V | |
| 8 | fvex 6869 | . . 3 ⊢ (((mRSubst‘𝑇)‘𝐹)‘(2nd ‘𝑋)) ∈ V | |
| 9 | 7, 8 | op1std 7969 | . 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 1095 = wceq 1554 ∈ wcel 2136 ⊆ wss 3899 〈cop 4582 ⟶wf 6506 ‘cfv 6510 1st c1st 7957 2nd c2nd 7958 mVRcmvar 35759 mRExcmrex 35764 mExcmex 35765 mRSubstcmrsub 35768 mSubstcmsub 35769 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1809 ax-4 1823 ax-5 1924 ax-6 1981 ax-7 2022 ax-8 2138 ax-9 2146 ax-10 2169 ax-11 2185 ax-12 2206 ax-ext 2728 ax-rep 5221 ax-sep 5240 ax-nul 5250 ax-pow 5316 ax-pr 5384 ax-un 7707 |
| This theorem depends on definitions: df-bi 209 df-an 399 df-or 857 df-3an 1097 df-tru 1557 df-fal 1567 df-ex 1794 df-nf 1798 df-sb 2085 df-mo 2560 df-eu 2590 df-clab 2735 df-cleq 2748 df-clel 2831 df-nfc 2905 df-ne 2952 df-ral 3071 df-rex 3081 df-reu 3362 df-rab 3409 df-v 3450 df-sbc 3740 df-csb 3848 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4281 df-if 4475 df-pw 4551 df-sn 4577 df-pr 4579 df-op 4583 df-uni 4860 df-iun 4945 df-br 5095 df-opab 5157 df-mpt 5176 df-id 5535 df-xp 5646 df-rel 5647 df-cnv 5648 df-co 5649 df-dm 5650 df-rn 5651 df-res 5652 df-ima 5653 df-iota 6466 df-fun 6512 df-fn 6513 df-f 6514 df-f1 6515 df-fo 6516 df-f1o 6517 df-fv 6518 df-ov 7388 df-oprab 7389 df-mpo 7390 df-1st 7959 df-pm 8799 df-msub 35789 |
| This theorem is referenced by: (None) |
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