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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 2752 | . . 3 ⊢ (mRSubst‘𝑇) = (mRSubst‘𝑇) | |
| 6 | 1, 2, 3, 4, 5 | msubval 35813 | . 2 ⊢ ((𝐹:𝐴⟶𝑅 ∧ 𝐴 ⊆ 𝑉 ∧ 𝑋 ∈ 𝐸) → ((𝑆‘𝐹)‘𝑋) = 〈(1st ‘𝑋), (((mRSubst‘𝑇)‘𝐹)‘(2nd ‘𝑋))〉) |
| 7 | fvex 6865 | . . 3 ⊢ (1st ‘𝑋) ∈ V | |
| 8 | fvex 6865 | . . 3 ⊢ (((mRSubst‘𝑇)‘𝐹)‘(2nd ‘𝑋)) ∈ V | |
| 9 | 7, 8 | op1std 7965 | . 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 1550 ∈ wcel 2132 ⊆ wss 3895 〈cop 4578 ⟶wf 6502 ‘cfv 6506 1st c1st 7953 2nd c2nd 7954 mVRcmvar 35749 mRExcmrex 35754 mExcmex 35755 mRSubstcmrsub 35758 mSubstcmsub 35759 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1805 ax-4 1819 ax-5 1920 ax-6 1977 ax-7 2018 ax-8 2134 ax-9 2142 ax-10 2165 ax-11 2181 ax-12 2202 ax-ext 2724 ax-rep 5217 ax-sep 5236 ax-nul 5246 ax-pow 5312 ax-pr 5380 ax-un 7703 |
| This theorem depends on definitions: df-bi 209 df-an 399 df-or 857 df-3an 1097 df-tru 1553 df-fal 1563 df-ex 1790 df-nf 1794 df-sb 2081 df-mo 2556 df-eu 2586 df-clab 2731 df-cleq 2744 df-clel 2827 df-nfc 2901 df-ne 2948 df-ral 3067 df-rex 3077 df-reu 3358 df-rab 3405 df-v 3446 df-sbc 3736 df-csb 3844 df-dif 3898 df-un 3900 df-in 3902 df-ss 3912 df-nul 4277 df-if 4471 df-pw 4547 df-sn 4573 df-pr 4575 df-op 4579 df-uni 4856 df-iun 4941 df-br 5091 df-opab 5153 df-mpt 5172 df-id 5531 df-xp 5642 df-rel 5643 df-cnv 5644 df-co 5645 df-dm 5646 df-rn 5647 df-res 5648 df-ima 5649 df-iota 6462 df-fun 6508 df-fn 6509 df-f 6510 df-f1 6511 df-fo 6512 df-f1o 6513 df-fv 6514 df-ov 7384 df-oprab 7385 df-mpo 7386 df-1st 7955 df-pm 8795 df-msub 35779 |
| This theorem is referenced by: (None) |
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