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 7863 | . . . . . . . . 9 ⊢ (𝑒 ∈ ((mTC‘𝑇) × 𝑅) → (1st ‘𝑒) ∈ (mTC‘𝑇)) | |
2 | eqid 2738 | . . . . . . . . . 10 ⊢ (mTC‘𝑇) = (mTC‘𝑇) | |
3 | msubff.e | . . . . . . . . . 10 ⊢ 𝐸 = (mEx‘𝑇) | |
4 | msubff.r | . . . . . . . . . 10 ⊢ 𝑅 = (mREx‘𝑇) | |
5 | 2, 3, 4 | mexval 33464 | . . . . . . . . 9 ⊢ 𝐸 = ((mTC‘𝑇) × 𝑅) |
6 | 1, 5 | eleq2s 2857 | . . . . . . . 8 ⊢ (𝑒 ∈ 𝐸 → (1st ‘𝑒) ∈ (mTC‘𝑇)) |
7 | 6 | adantl 482 | . . . . . . 7 ⊢ (((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) ∧ 𝑒 ∈ 𝐸) → (1st ‘𝑒) ∈ (mTC‘𝑇)) |
8 | msubff.v | . . . . . . . . . . 11 ⊢ 𝑉 = (mVR‘𝑇) | |
9 | eqid 2738 | . . . . . . . . . . 11 ⊢ (mRSubst‘𝑇) = (mRSubst‘𝑇) | |
10 | 8, 4, 9 | mrsubff 33474 | . . . . . . . . . 10 ⊢ (𝑇 ∈ 𝑊 → (mRSubst‘𝑇):(𝑅 ↑pm 𝑉)⟶(𝑅 ↑m 𝑅)) |
11 | 10 | ffvelrnda 6961 | . . . . . . . . 9 ⊢ ((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → ((mRSubst‘𝑇)‘𝑓) ∈ (𝑅 ↑m 𝑅)) |
12 | elmapi 8637 | . . . . . . . . 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 7864 | . . . . . . . . 9 ⊢ (𝑒 ∈ ((mTC‘𝑇) × 𝑅) → (2nd ‘𝑒) ∈ 𝑅) | |
15 | 14, 5 | eleq2s 2857 | . . . . . . . 8 ⊢ (𝑒 ∈ 𝐸 → (2nd ‘𝑒) ∈ 𝑅) |
16 | ffvelrn 6959 | . . . . . . . 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 5625 | . . . . . . 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 2850 | . . . . 5 ⊢ (((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) ∧ 𝑒 ∈ 𝐸) → 〈(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒))〉 ∈ 𝐸) |
21 | 20 | fmpttd 6989 | . . . 4 ⊢ ((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → (𝑒 ∈ 𝐸 ↦ 〈(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒))〉):𝐸⟶𝐸) |
22 | 3 | fvexi 6788 | . . . . 5 ⊢ 𝐸 ∈ V |
23 | 22, 22 | elmap 8659 | . . . 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 233 | . . 3 ⊢ ((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → (𝑒 ∈ 𝐸 ↦ 〈(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒))〉) ∈ (𝐸 ↑m 𝐸)) |
25 | 24 | fmpttd 6989 | . 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 33485 | . . 3 ⊢ (𝑇 ∈ 𝑊 → 𝑆 = (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝐸 ↦ 〈(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒))〉))) |
28 | 27 | feq1d 6585 | . 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 256 | 1 ⊢ (𝑇 ∈ 𝑊 → 𝑆:(𝑅 ↑pm 𝑉)⟶(𝐸 ↑m 𝐸)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 〈cop 4567 ↦ cmpt 5157 × cxp 5587 ⟶wf 6429 ‘cfv 6433 (class class class)co 7275 1st c1st 7829 2nd c2nd 7830 ↑m cmap 8615 ↑pm cpm 8616 mVRcmvar 33423 mTCcmtc 33426 mRExcmrex 33428 mExcmex 33429 mRSubstcmrsub 33432 mSubstcmsub 33433 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-int 4880 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-1o 8297 df-er 8498 df-map 8617 df-pm 8618 df-en 8734 df-dom 8735 df-sdom 8736 df-fin 8737 df-card 9697 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-2 12036 df-n0 12234 df-z 12320 df-uz 12583 df-fz 13240 df-fzo 13383 df-seq 13722 df-hash 14045 df-word 14218 df-concat 14274 df-s1 14301 df-struct 16848 df-sets 16865 df-slot 16883 df-ndx 16895 df-base 16913 df-ress 16942 df-plusg 16975 df-0g 17152 df-gsum 17153 df-mgm 18326 df-sgrp 18375 df-mnd 18386 df-submnd 18431 df-frmd 18488 df-mrex 33448 df-mex 33449 df-mrsub 33452 df-msub 33453 |
This theorem is referenced by: msubf 33494 msubff1 33518 mclsind 33532 |
Copyright terms: Public domain | W3C validator |