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Mathbox for Mario Carneiro |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > msubff1o | 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 |
---|---|
msubff1.v | ⊢ 𝑉 = (mVR‘𝑇) |
msubff1.r | ⊢ 𝑅 = (mREx‘𝑇) |
msubff1.s | ⊢ 𝑆 = (mSubst‘𝑇) |
Ref | Expression |
---|---|
msubff1o | ⊢ (𝑇 ∈ mFS → (𝑆 ↾ (𝑅 ↑m 𝑉)):(𝑅 ↑m 𝑉)–1-1-onto→ran 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | msubff1.v | . . . 4 ⊢ 𝑉 = (mVR‘𝑇) | |
2 | msubff1.r | . . . 4 ⊢ 𝑅 = (mREx‘𝑇) | |
3 | msubff1.s | . . . 4 ⊢ 𝑆 = (mSubst‘𝑇) | |
4 | eqid 2725 | . . . 4 ⊢ (mEx‘𝑇) = (mEx‘𝑇) | |
5 | 1, 2, 3, 4 | msubff1 35297 | . . 3 ⊢ (𝑇 ∈ mFS → (𝑆 ↾ (𝑅 ↑m 𝑉)):(𝑅 ↑m 𝑉)–1-1→((mEx‘𝑇) ↑m (mEx‘𝑇))) |
6 | f1f1orn 6849 | . . 3 ⊢ ((𝑆 ↾ (𝑅 ↑m 𝑉)):(𝑅 ↑m 𝑉)–1-1→((mEx‘𝑇) ↑m (mEx‘𝑇)) → (𝑆 ↾ (𝑅 ↑m 𝑉)):(𝑅 ↑m 𝑉)–1-1-onto→ran (𝑆 ↾ (𝑅 ↑m 𝑉))) | |
7 | 5, 6 | syl 17 | . 2 ⊢ (𝑇 ∈ mFS → (𝑆 ↾ (𝑅 ↑m 𝑉)):(𝑅 ↑m 𝑉)–1-1-onto→ran (𝑆 ↾ (𝑅 ↑m 𝑉))) |
8 | 1, 2, 3 | msubrn 35270 | . . . 4 ⊢ ran 𝑆 = (𝑆 “ (𝑅 ↑m 𝑉)) |
9 | df-ima 5691 | . . . 4 ⊢ (𝑆 “ (𝑅 ↑m 𝑉)) = ran (𝑆 ↾ (𝑅 ↑m 𝑉)) | |
10 | 8, 9 | eqtri 2753 | . . 3 ⊢ ran 𝑆 = ran (𝑆 ↾ (𝑅 ↑m 𝑉)) |
11 | f1oeq3 6828 | . . 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)))) | |
12 | 10, 11 | 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))) |
13 | 7, 12 | sylibr 233 | 1 ⊢ (𝑇 ∈ mFS → (𝑆 ↾ (𝑅 ↑m 𝑉)):(𝑅 ↑m 𝑉)–1-1-onto→ran 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1533 ∈ wcel 2098 ran crn 5679 ↾ cres 5680 “ cima 5681 –1-1→wf1 6546 –1-1-onto→wf1o 6548 ‘cfv 6549 (class class class)co 7419 ↑m cmap 8845 mVRcmvar 35202 mRExcmrex 35207 mExcmex 35208 mSubstcmsub 35212 mFScmfs 35217 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-1cn 11198 ax-icn 11199 ax-addcl 11200 ax-addrcl 11201 ax-mulcl 11202 ax-mulrcl 11203 ax-mulcom 11204 ax-addass 11205 ax-mulass 11206 ax-distr 11207 ax-i2m1 11208 ax-1ne0 11209 ax-1rid 11210 ax-rnegex 11211 ax-rrecex 11212 ax-cnre 11213 ax-pre-lttri 11214 ax-pre-lttrn 11215 ax-pre-ltadd 11216 ax-pre-mulgt0 11217 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rmo 3363 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-int 4951 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-pred 6307 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-riota 7375 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-frecs 8287 df-wrecs 8318 df-recs 8392 df-rdg 8431 df-1o 8487 df-er 8725 df-map 8847 df-pm 8848 df-en 8965 df-dom 8966 df-sdom 8967 df-fin 8968 df-card 9964 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-sub 11478 df-neg 11479 df-nn 12246 df-2 12308 df-n0 12506 df-z 12592 df-uz 12856 df-fz 13520 df-fzo 13663 df-seq 14003 df-hash 14326 df-word 14501 df-concat 14557 df-s1 14582 df-struct 17119 df-sets 17136 df-slot 17154 df-ndx 17166 df-base 17184 df-ress 17213 df-plusg 17249 df-0g 17426 df-gsum 17427 df-mgm 18603 df-sgrp 18682 df-mnd 18698 df-submnd 18744 df-frmd 18809 df-mrex 35227 df-mex 35228 df-mrsub 35231 df-msub 35232 df-mfs 35237 |
This theorem is referenced by: (None) |
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