Mathbox for Mario Carneiro |
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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 7713 | . . . . . . . . 9 ⊢ (𝑒 ∈ ((mTC‘𝑇) × 𝑅) → (1st ‘𝑒) ∈ (mTC‘𝑇)) | |
2 | eqid 2819 | . . . . . . . . . 10 ⊢ (mTC‘𝑇) = (mTC‘𝑇) | |
3 | msubff.e | . . . . . . . . . 10 ⊢ 𝐸 = (mEx‘𝑇) | |
4 | msubff.r | . . . . . . . . . 10 ⊢ 𝑅 = (mREx‘𝑇) | |
5 | 2, 3, 4 | mexval 32742 | . . . . . . . . 9 ⊢ 𝐸 = ((mTC‘𝑇) × 𝑅) |
6 | 1, 5 | eleq2s 2929 | . . . . . . . 8 ⊢ (𝑒 ∈ 𝐸 → (1st ‘𝑒) ∈ (mTC‘𝑇)) |
7 | 6 | adantl 484 | . . . . . . 7 ⊢ (((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) ∧ 𝑒 ∈ 𝐸) → (1st ‘𝑒) ∈ (mTC‘𝑇)) |
8 | msubff.v | . . . . . . . . . . 11 ⊢ 𝑉 = (mVR‘𝑇) | |
9 | eqid 2819 | . . . . . . . . . . 11 ⊢ (mRSubst‘𝑇) = (mRSubst‘𝑇) | |
10 | 8, 4, 9 | mrsubff 32752 | . . . . . . . . . 10 ⊢ (𝑇 ∈ 𝑊 → (mRSubst‘𝑇):(𝑅 ↑pm 𝑉)⟶(𝑅 ↑m 𝑅)) |
11 | 10 | ffvelrnda 6844 | . . . . . . . . 9 ⊢ ((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → ((mRSubst‘𝑇)‘𝑓) ∈ (𝑅 ↑m 𝑅)) |
12 | elmapi 8420 | . . . . . . . . 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 7714 | . . . . . . . . 9 ⊢ (𝑒 ∈ ((mTC‘𝑇) × 𝑅) → (2nd ‘𝑒) ∈ 𝑅) | |
15 | 14, 5 | eleq2s 2929 | . . . . . . . 8 ⊢ (𝑒 ∈ 𝐸 → (2nd ‘𝑒) ∈ 𝑅) |
16 | ffvelrn 6842 | . . . . . . . 8 ⊢ ((((mRSubst‘𝑇)‘𝑓):𝑅⟶𝑅 ∧ (2nd ‘𝑒) ∈ 𝑅) → (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒)) ∈ 𝑅) | |
17 | 13, 15, 16 | syl2an 597 | . . . . . . 7 ⊢ (((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) ∧ 𝑒 ∈ 𝐸) → (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒)) ∈ 𝑅) |
18 | opelxp 5584 | . . . . . . 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 585 | . . . . . 6 ⊢ (((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) ∧ 𝑒 ∈ 𝐸) → 〈(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒))〉 ∈ ((mTC‘𝑇) × 𝑅)) |
20 | 19, 5 | eleqtrrdi 2922 | . . . . 5 ⊢ (((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) ∧ 𝑒 ∈ 𝐸) → 〈(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒))〉 ∈ 𝐸) |
21 | 20 | fmpttd 6872 | . . . 4 ⊢ ((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → (𝑒 ∈ 𝐸 ↦ 〈(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒))〉):𝐸⟶𝐸) |
22 | 3 | fvexi 6677 | . . . . 5 ⊢ 𝐸 ∈ V |
23 | 22, 22 | elmap 8427 | . . . 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 236 | . . 3 ⊢ ((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → (𝑒 ∈ 𝐸 ↦ 〈(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒))〉) ∈ (𝐸 ↑m 𝐸)) |
25 | 24 | fmpttd 6872 | . 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 32763 | . . 3 ⊢ (𝑇 ∈ 𝑊 → 𝑆 = (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝐸 ↦ 〈(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒))〉))) |
28 | 27 | feq1d 6492 | . 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 259 | 1 ⊢ (𝑇 ∈ 𝑊 → 𝑆:(𝑅 ↑pm 𝑉)⟶(𝐸 ↑m 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1531 ∈ wcel 2108 〈cop 4565 ↦ cmpt 5137 × cxp 5546 ⟶wf 6344 ‘cfv 6348 (class class class)co 7148 1st c1st 7679 2nd c2nd 7680 ↑m cmap 8398 ↑pm cpm 8399 mVRcmvar 32701 mTCcmtc 32704 mRExcmrex 32706 mExcmex 32707 mRSubstcmrsub 32710 mSubstcmsub 32711 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1905 ax-6 1964 ax-7 2009 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2154 ax-12 2170 ax-ext 2791 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7453 ax-cnex 10585 ax-resscn 10586 ax-1cn 10587 ax-icn 10588 ax-addcl 10589 ax-addrcl 10590 ax-mulcl 10591 ax-mulrcl 10592 ax-mulcom 10593 ax-addass 10594 ax-mulass 10595 ax-distr 10596 ax-i2m1 10597 ax-1ne0 10598 ax-1rid 10599 ax-rnegex 10600 ax-rrecex 10601 ax-cnre 10602 ax-pre-lttri 10603 ax-pre-lttrn 10604 ax-pre-ltadd 10605 ax-pre-mulgt0 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1534 df-ex 1775 df-nf 1779 df-sb 2064 df-mo 2616 df-eu 2648 df-clab 2798 df-cleq 2812 df-clel 2891 df-nfc 2961 df-ne 3015 df-nel 3122 df-ral 3141 df-rex 3142 df-reu 3143 df-rmo 3144 df-rab 3145 df-v 3495 df-sbc 3771 df-csb 3882 df-dif 3937 df-un 3939 df-in 3941 df-ss 3950 df-pss 3952 df-nul 4290 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 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-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 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-riota 7106 df-ov 7151 df-oprab 7152 df-mpo 7153 df-om 7573 df-1st 7681 df-2nd 7682 df-wrecs 7939 df-recs 8000 df-rdg 8038 df-1o 8094 df-oadd 8098 df-er 8281 df-map 8400 df-pm 8401 df-en 8502 df-dom 8503 df-sdom 8504 df-fin 8505 df-card 9360 df-pnf 10669 df-mnf 10670 df-xr 10671 df-ltxr 10672 df-le 10673 df-sub 10864 df-neg 10865 df-nn 11631 df-2 11692 df-n0 11890 df-z 11974 df-uz 12236 df-fz 12885 df-fzo 13026 df-seq 13362 df-hash 13683 df-word 13854 df-concat 13915 df-s1 13942 df-struct 16477 df-ndx 16478 df-slot 16479 df-base 16481 df-sets 16482 df-ress 16483 df-plusg 16570 df-0g 16707 df-gsum 16708 df-mgm 17844 df-sgrp 17893 df-mnd 17904 df-submnd 17949 df-frmd 18006 df-mrex 32726 df-mex 32727 df-mrsub 32730 df-msub 32731 |
This theorem is referenced by: msubf 32772 msubff1 32796 mclsind 32810 |
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