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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 8045 | . . . . . . . . 9 ⊢ (𝑒 ∈ ((mTC‘𝑇) × 𝑅) → (1st ‘𝑒) ∈ (mTC‘𝑇)) | |
2 | eqid 2735 | . . . . . . . . . 10 ⊢ (mTC‘𝑇) = (mTC‘𝑇) | |
3 | msubff.e | . . . . . . . . . 10 ⊢ 𝐸 = (mEx‘𝑇) | |
4 | msubff.r | . . . . . . . . . 10 ⊢ 𝑅 = (mREx‘𝑇) | |
5 | 2, 3, 4 | mexval 35487 | . . . . . . . . 9 ⊢ 𝐸 = ((mTC‘𝑇) × 𝑅) |
6 | 1, 5 | eleq2s 2857 | . . . . . . . 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 2735 | . . . . . . . . . . 11 ⊢ (mRSubst‘𝑇) = (mRSubst‘𝑇) | |
10 | 8, 4, 9 | mrsubff 35497 | . . . . . . . . . 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 8888 | . . . . . . . . 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 8046 | . . . . . . . . 9 ⊢ (𝑒 ∈ ((mTC‘𝑇) × 𝑅) → (2nd ‘𝑒) ∈ 𝑅) | |
15 | 14, 5 | eleq2s 2857 | . . . . . . . 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 5725 | . . . . . . 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 7135 | . . . 4 ⊢ ((𝑇 ∈ 𝑊 ∧ 𝑓 ∈ (𝑅 ↑pm 𝑉)) → (𝑒 ∈ 𝐸 ↦ 〈(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒))〉):𝐸⟶𝐸) |
22 | 3 | fvexi 6921 | . . . . 5 ⊢ 𝐸 ∈ V |
23 | 22, 22 | elmap 8910 | . . . 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 35508 | . . 3 ⊢ (𝑇 ∈ 𝑊 → 𝑆 = (𝑓 ∈ (𝑅 ↑pm 𝑉) ↦ (𝑒 ∈ 𝐸 ↦ 〈(1st ‘𝑒), (((mRSubst‘𝑇)‘𝑓)‘(2nd ‘𝑒))〉))) |
28 | 27 | feq1d 6721 | . 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 1537 ∈ wcel 2106 〈cop 4637 ↦ cmpt 5231 × cxp 5687 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 1st c1st 8011 2nd c2nd 8012 ↑m cmap 8865 ↑pm cpm 8866 mVRcmvar 35446 mTCcmtc 35449 mRExcmrex 35451 mExcmex 35452 mRSubstcmrsub 35455 mSubstcmsub 35456 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 ax-pre-mulgt0 11230 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-pm 8868 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-card 9977 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-sub 11492 df-neg 11493 df-nn 12265 df-2 12327 df-n0 12525 df-z 12612 df-uz 12877 df-fz 13545 df-fzo 13692 df-seq 14040 df-hash 14367 df-word 14550 df-concat 14606 df-s1 14631 df-struct 17181 df-sets 17198 df-slot 17216 df-ndx 17228 df-base 17246 df-ress 17275 df-plusg 17311 df-0g 17488 df-gsum 17489 df-mgm 18666 df-sgrp 18745 df-mnd 18761 df-submnd 18810 df-frmd 18875 df-mrex 35471 df-mex 35472 df-mrsub 35475 df-msub 35476 |
This theorem is referenced by: msubf 35517 msubff1 35541 mclsind 35555 |
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