| Mathbox for Mario Carneiro |
< Previous
Next >
Nearby theorems |
||
| Mirrors > Home > MPE Home > Th. List > Mathboxes > msubff | Structured version Visualization version GIF version | ||
| Description: A substitution is a function from 𝐸 to 𝐸. (Contributed by Mario Carneiro, 18-Jul-2016.) |
| Ref | Expression |
|---|---|
| msubff.v | ⊢ 𝑉 = (mVR‘𝑇) |
| msubff.r | ⊢ 𝑅 = (mREx‘𝑇) |
| msubff.s | ⊢ 𝑆 = (mSubst‘𝑇) |
| msubff.e | ⊢ 𝐸 = (mEx‘𝑇) |
| Ref | Expression |
|---|---|
| msubff | ⊢ (𝑇 ∈ 𝑊 → 𝑆:(𝑅 ↑pm 𝑉)⟶(𝐸 ↑m 𝐸)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | xp1st 8046 | . . . . . . . . 9 ⊢ (𝑒 ∈ ((mTC‘𝑇) × 𝑅) → (1st ‘𝑒) ∈ (mTC‘𝑇)) | |
| 2 | eqid 2737 | . . . . . . . . . 10 ⊢ (mTC‘𝑇) = (mTC‘𝑇) | |
| 3 | msubff.e | . . . . . . . . . 10 ⊢ 𝐸 = (mEx‘𝑇) | |
| 4 | msubff.r | . . . . . . . . . 10 ⊢ 𝑅 = (mREx‘𝑇) | |
| 5 | 2, 3, 4 | mexval 35507 | . . . . . . . . 9 ⊢ 𝐸 = ((mTC‘𝑇) × 𝑅) |
| 6 | 1, 5 | eleq2s 2859 | . . . . . . . 8 ⊢ (𝑒 ∈ 𝐸 → (1st ‘𝑒) ∈ (mTC‘𝑇)) |
| 7 | 6 | adantl 481 | . . . . . . 7 ⊢ (((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) ∧ 𝑒 ∈ 𝐸) → (1st ‘𝑒) ∈ (mTC‘𝑇)) |
| 8 | msubff.v | . . . . . . . . . . 11 ⊢ 𝑉 = (mVR‘𝑇) | |
| 9 | eqid 2737 | . . . . . . . . . . 11 ⊢ (mRSubst‘𝑇) = (mRSubst‘𝑇) | |
| 10 | 8, 4, 9 | mrsubff 35517 | . . . . . . . . . 10 ⊢ (𝑇 ∈ 𝑊 → (mRSubst‘𝑇):(𝑅 ↑pm 𝑉)⟶(𝑅 ↑m 𝑅)) |
| 11 | 10 | ffvelcdmda 7104 | . . . . . . . . 9 ⊢ ((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → ((mRSubst‘𝑇)‘𝑓) ∈ (𝑅 ↑m 𝑅)) |
| 12 | elmapi 8889 | . . . . . . . . 9 ⊢ (((mRSubst‘𝑇)‘𝑓) ∈ (𝑅 ↑m 𝑅) → ((mRSubst‘𝑇)‘𝑓):𝑅⟶𝑅) | |
| 13 | 11, 12 | syl 17 | . . . . . . . 8 ⊢ ((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → ((mRSubst‘𝑇)‘𝑓):𝑅⟶𝑅) |
| 14 | xp2nd 8047 | . . . . . . . . 9 ⊢ (𝑒 ∈ ((mTC‘𝑇) × 𝑅) → (2nd ‘𝑒) ∈ 𝑅) | |
| 15 | 14, 5 | eleq2s 2859 | . . . . . . . 8 ⊢ (𝑒 ∈ 𝐸 → (2nd ‘𝑒) ∈ 𝑅) |
| 16 | ffvelcdm 7101 | . . . . . . . 8 ⊢ ((((mRSubst‘𝑇)‘𝑓):𝑅⟶𝑅 ∧ (2nd ‘𝑒) ∈ 𝑅) → (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒)) ∈ 𝑅) | |
| 17 | 13, 15, 16 | syl2an 596 | . . . . . . 7 ⊢ (((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) ∧ 𝑒 ∈ 𝐸) → (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒)) ∈ 𝑅) |
| 18 | opelxp 5721 | . . . . . . 7 ⊢ (〈(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒))〉 ∈ ((mTC‘𝑇) × 𝑅) ↔ ((1st ‘𝑒) ∈ (mTC‘𝑇) ∧ (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒)) ∈ 𝑅)) | |
| 19 | 7, 17, 18 | sylanbrc 583 | . . . . . 6 ⊢ (((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) ∧ 𝑒 ∈ 𝐸) → 〈(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒))〉 ∈ ((mTC‘𝑇) × 𝑅)) |
| 20 | 19, 5 | eleqtrrdi 2852 | . . . . 5 ⊢ (((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) ∧ 𝑒 ∈ 𝐸) → 〈(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒))〉 ∈ 𝐸) |
| 21 | 20 | fmpttd 7135 | . . . 4 ⊢ ((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → (𝑒 ∈ 𝐸 ↦ 〈(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒))〉):𝐸⟶𝐸) |
| 22 | 3 | fvexi 6920 | . . . . 5 ⊢ 𝐸 ∈ V |
| 23 | 22, 22 | elmap 8911 | . . . 4 ⊢ ((𝑒 ∈ 𝐸 ↦ 〈(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒))〉) ∈ (𝐸 ↑m 𝐸) ↔ (𝑒 ∈ 𝐸 ↦ 〈(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒))〉):𝐸⟶𝐸) |
| 24 | 21, 23 | sylibr 234 | . . 3 ⊢ ((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → (𝑒 ∈ 𝐸 ↦ 〈(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒))〉) ∈ (𝐸 ↑m 𝐸)) |
| 25 | 24 | fmpttd 7135 | . 2 ⊢ (𝑇 ∈ 𝑊 → (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝐸 ↦ 〈(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒))〉)):(𝑅 ↑pm 𝑉)⟶(𝐸 ↑m 𝐸)) |
| 26 | msubff.s | . . . 4 ⊢ 𝑆 = (mSubst‘𝑇) | |
| 27 | 8, 4, 26, 3, 9 | msubffval 35528 | . . 3 ⊢ (𝑇 ∈ 𝑊 → 𝑆 = (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝐸 ↦ 〈(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒))〉))) |
| 28 | 27 | feq1d 6720 | . 2 ⊢ (𝑇 ∈ 𝑊 → (𝑆:(𝑅 ↑pm 𝑉)⟶(𝐸 ↑m 𝐸) ↔ (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝐸 ↦ 〈(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒))〉)):(𝑅 ↑pm 𝑉)⟶(𝐸 ↑m 𝐸))) |
| 29 | 25, 28 | mpbird 257 | 1 ⊢ (𝑇 ∈ 𝑊 → 𝑆:(𝑅 ↑pm 𝑉)⟶(𝐸 ↑m 𝐸)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 〈cop 4632 ↦ cmpt 5225 × cxp 5683 ⟶wf 6557 ‘cfv 6561 (class class class)co 7431 1st c1st 8012 2nd c2nd 8013 ↑m cmap 8866 ↑pm cpm 8867 mVRcmvar 35466 mTCcmtc 35469 mRExcmrex 35471 mExcmex 35472 mRSubstcmrsub 35475 mSubstcmsub 35476 |
| 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 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-pm 8869 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-2 12329 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 df-seq 14043 df-hash 14370 df-word 14553 df-concat 14609 df-s1 14634 df-struct 17184 df-sets 17201 df-slot 17219 df-ndx 17231 df-base 17248 df-ress 17275 df-plusg 17310 df-0g 17486 df-gsum 17487 df-mgm 18653 df-sgrp 18732 df-mnd 18748 df-submnd 18797 df-frmd 18862 df-mrex 35491 df-mex 35492 df-mrsub 35495 df-msub 35496 |
| This theorem is referenced by: msubf 35537 msubff1 35561 mclsind 35575 |
| Copyright terms: Public domain | W3C validator |