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Mirrors > Home > MPE Home > Th. List > Mathboxes > msubrsub | Structured version Visualization version GIF version |
Description: A substitution applied to an expression. (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) |
msubffval.o | ⊢ 𝑂 = (mRSubst‘𝑇) |
Ref | Expression |
---|---|
msubrsub | ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) → (2nd ‘((𝑆‘𝐹)‘𝑋)) = ((𝑂‘𝐹)‘(2nd ‘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 | msubffval.o | . . 3 ⊢ 𝑂 = (mRSubst‘𝑇) | |
6 | 1, 2, 3, 4, 5 | msubval 35508 | . 2 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) → ((𝑆‘𝐹)‘𝑋) = 〈(1st ‘𝑋), ((𝑂‘𝐹)‘(2nd ‘𝑋))〉) |
7 | fvex 6917 | . . 3 ⊢ (1st ‘𝑋) ∈ V | |
8 | fvex 6917 | . . 3 ⊢ ((𝑂‘𝐹)‘(2nd ‘𝑋)) ∈ V | |
9 | 7, 8 | op2ndd 8021 | . 2 ⊢ (((𝑆‘𝐹)‘𝑋) = 〈(1st ‘𝑋), ((𝑂‘𝐹)‘(2nd ‘𝑋))〉 → (2nd ‘((𝑆‘𝐹)‘𝑋)) = ((𝑂‘𝐹)‘(2nd ‘𝑋))) |
10 | 6, 9 | syl 17 | 1 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) → (2nd ‘((𝑆‘𝐹)‘𝑋)) = ((𝑂‘𝐹)‘(2nd ‘𝑋))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ⊆ wss 3950 〈cop 4630 ⟶wf 6555 ‘cfv 6559 1st c1st 8008 2nd c2nd 8009 mVRcmvar 35444 mRExcmrex 35449 mExcmex 35450 mRSubstcmrsub 35453 mSubstcmsub 35454 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5277 ax-sep 5294 ax-nul 5304 ax-pow 5363 ax-pr 5430 ax-un 7751 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4906 df-iun 4991 df-br 5142 df-opab 5204 df-mpt 5224 df-id 5576 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-iota 6512 df-fun 6561 df-fn 6562 df-f 6563 df-f1 6564 df-fo 6565 df-f1o 6566 df-fv 6567 df-ov 7432 df-oprab 7433 df-mpo 7434 df-2nd 8011 df-pm 8865 df-msub 35474 |
This theorem is referenced by: (None) |
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