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Mirrors > Home > MPE Home > Th. List > Mathboxes > mrsubff1o | Structured version Visualization version GIF version |
Description: When restricted to complete mappings, the substitution-producing function is bijective to the set of all substitutions. (Contributed by Mario Carneiro, 18-Jul-2016.) |
Ref | Expression |
---|---|
mrsubvr.v | ⊢ 𝑉 = (mVR‘𝑇) |
mrsubvr.r | ⊢ 𝑅 = (mREx‘𝑇) |
mrsubvr.s | ⊢ 𝑆 = (mRSubst‘𝑇) |
Ref | Expression |
---|---|
mrsubff1o | ⊢ (𝑇 ∈ 𝑊 → (𝑆 ↾ (𝑅 ↑m 𝑉)):(𝑅 ↑m 𝑉)–1-1-onto→ran 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mrsubvr.v | . . . 4 ⊢ 𝑉 = (mVR‘𝑇) | |
2 | mrsubvr.r | . . . 4 ⊢ 𝑅 = (mREx‘𝑇) | |
3 | mrsubvr.s | . . . 4 ⊢ 𝑆 = (mRSubst‘𝑇) | |
4 | 1, 2, 3 | mrsubff1 33161 | . . 3 ⊢ (𝑇 ∈ 𝑊 → (𝑆 ↾ (𝑅 ↑m 𝑉)):(𝑅 ↑m 𝑉)–1-1→(𝑅 ↑m 𝑅)) |
5 | f1f1orn 6661 | . . 3 ⊢ ((𝑆 ↾ (𝑅 ↑m 𝑉)):(𝑅 ↑m 𝑉)–1-1→(𝑅 ↑m 𝑅) → (𝑆 ↾ (𝑅 ↑m 𝑉)):(𝑅 ↑m 𝑉)–1-1-onto→ran (𝑆 ↾ (𝑅 ↑m 𝑉))) | |
6 | 4, 5 | syl 17 | . 2 ⊢ (𝑇 ∈ 𝑊 → (𝑆 ↾ (𝑅 ↑m 𝑉)):(𝑅 ↑m 𝑉)–1-1-onto→ran (𝑆 ↾ (𝑅 ↑m 𝑉))) |
7 | 1, 2, 3 | mrsubrn 33160 | . . . 4 ⊢ ran 𝑆 = (𝑆 “ (𝑅 ↑m 𝑉)) |
8 | df-ima 5553 | . . . 4 ⊢ (𝑆 “ (𝑅 ↑m 𝑉)) = ran (𝑆 ↾ (𝑅 ↑m 𝑉)) | |
9 | 7, 8 | eqtri 2762 | . . 3 ⊢ ran 𝑆 = ran (𝑆 ↾ (𝑅 ↑m 𝑉)) |
10 | f1oeq3 6640 | . . 3 ⊢ (ran 𝑆 = ran (𝑆 ↾ (𝑅 ↑m 𝑉)) → ((𝑆 ↾ (𝑅 ↑m 𝑉)):(𝑅 ↑m 𝑉)–1-1-onto→ran 𝑆 ↔ (𝑆 ↾ (𝑅 ↑m 𝑉)):(𝑅 ↑m 𝑉)–1-1-onto→ran (𝑆 ↾ (𝑅 ↑m 𝑉)))) | |
11 | 9, 10 | ax-mp 5 | . 2 ⊢ ((𝑆 ↾ (𝑅 ↑m 𝑉)):(𝑅 ↑m 𝑉)–1-1-onto→ran 𝑆 ↔ (𝑆 ↾ (𝑅 ↑m 𝑉)):(𝑅 ↑m 𝑉)–1-1-onto→ran (𝑆 ↾ (𝑅 ↑m 𝑉))) |
12 | 6, 11 | sylibr 237 | 1 ⊢ (𝑇 ∈ 𝑊 → (𝑆 ↾ (𝑅 ↑m 𝑉)):(𝑅 ↑m 𝑉)–1-1-onto→ran 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1543 ∈ wcel 2110 ran crn 5541 ↾ cres 5542 “ cima 5543 –1-1→wf1 6366 –1-1-onto→wf1o 6368 ‘cfv 6369 (class class class)co 7202 ↑m cmap 8497 mVRcmvar 33108 mRExcmrex 33113 mRSubstcmrsub 33117 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2706 ax-rep 5168 ax-sep 5181 ax-nul 5188 ax-pow 5247 ax-pr 5311 ax-un 7512 ax-cnex 10768 ax-resscn 10769 ax-1cn 10770 ax-icn 10771 ax-addcl 10772 ax-addrcl 10773 ax-mulcl 10774 ax-mulrcl 10775 ax-mulcom 10776 ax-addass 10777 ax-mulass 10778 ax-distr 10779 ax-i2m1 10780 ax-1ne0 10781 ax-1rid 10782 ax-rnegex 10783 ax-rrecex 10784 ax-cnre 10785 ax-pre-lttri 10786 ax-pre-lttrn 10787 ax-pre-ltadd 10788 ax-pre-mulgt0 10789 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2537 df-eu 2566 df-clab 2713 df-cleq 2726 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rmo 3062 df-rab 3063 df-v 3403 df-sbc 3688 df-csb 3803 df-dif 3860 df-un 3862 df-in 3864 df-ss 3874 df-pss 3876 df-nul 4228 df-if 4430 df-pw 4505 df-sn 4532 df-pr 4534 df-tp 4536 df-op 4538 df-uni 4810 df-int 4850 df-iun 4896 df-br 5044 df-opab 5106 df-mpt 5125 df-tr 5151 df-id 5444 df-eprel 5449 df-po 5457 df-so 5458 df-fr 5498 df-we 5500 df-xp 5546 df-rel 5547 df-cnv 5548 df-co 5549 df-dm 5550 df-rn 5551 df-res 5552 df-ima 5553 df-pred 6149 df-ord 6205 df-on 6206 df-lim 6207 df-suc 6208 df-iota 6327 df-fun 6371 df-fn 6372 df-f 6373 df-f1 6374 df-fo 6375 df-f1o 6376 df-fv 6377 df-riota 7159 df-ov 7205 df-oprab 7206 df-mpo 7207 df-om 7634 df-1st 7750 df-2nd 7751 df-wrecs 8036 df-recs 8097 df-rdg 8135 df-1o 8191 df-er 8380 df-map 8499 df-pm 8500 df-en 8616 df-dom 8617 df-sdom 8618 df-fin 8619 df-card 9538 df-pnf 10852 df-mnf 10853 df-xr 10854 df-ltxr 10855 df-le 10856 df-sub 11047 df-neg 11048 df-nn 11814 df-2 11876 df-n0 12074 df-z 12160 df-uz 12422 df-fz 13079 df-fzo 13222 df-seq 13558 df-hash 13880 df-word 14053 df-concat 14109 df-s1 14136 df-struct 16686 df-ndx 16687 df-slot 16688 df-base 16690 df-sets 16691 df-ress 16692 df-plusg 16780 df-0g 16918 df-gsum 16919 df-mgm 18086 df-sgrp 18135 df-mnd 18146 df-submnd 18191 df-frmd 18248 df-mrex 33133 df-mrsub 33137 |
This theorem is referenced by: (None) |
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