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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 2610 | . . . 4 ⊢ (mEx‘𝑇) = (mEx‘𝑇) | |
5 | 1, 2, 3, 4 | msubff1 30707 | . . 3 ⊢ (𝑇 ∈ mFS → (𝑆 ↾ (𝑅 ↑𝑚 𝑉)):(𝑅 ↑𝑚 𝑉)–1-1→((mEx‘𝑇) ↑𝑚 (mEx‘𝑇))) |
6 | f1f1orn 6061 | . . 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 30680 | . . . 4 ⊢ ran 𝑆 = (𝑆 “ (𝑅 ↑𝑚 𝑉)) |
9 | df-ima 5051 | . . . 4 ⊢ (𝑆 “ (𝑅 ↑𝑚 𝑉)) = ran (𝑆 ↾ (𝑅 ↑𝑚 𝑉)) | |
10 | 8, 9 | eqtri 2632 | . . 3 ⊢ ran 𝑆 = ran (𝑆 ↾ (𝑅 ↑𝑚 𝑉)) |
11 | f1oeq3 6042 | . . 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 223 | 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 195 = wceq 1475 ∈ wcel 1977 ran crn 5039 ↾ cres 5040 “ cima 5041 –1-1→wf1 5801 –1-1-onto→wf1o 5803 ‘cfv 5804 (class class class)co 6549 ↑𝑚 cmap 7744 mVRcmvar 30612 mRExcmrex 30617 mExcmex 30618 mSubstcmsub 30622 mFScmfs 30627 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-1st 7059 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-oadd 7451 df-er 7629 df-map 7746 df-pm 7747 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-2 10956 df-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-seq 12664 df-hash 12980 df-word 13154 df-concat 13156 df-s1 13157 df-struct 15697 df-ndx 15698 df-slot 15699 df-base 15700 df-sets 15701 df-ress 15702 df-plusg 15781 df-0g 15925 df-gsum 15926 df-mgm 17065 df-sgrp 17107 df-mnd 17118 df-submnd 17159 df-frmd 17209 df-mrex 30637 df-mex 30638 df-mrsub 30641 df-msub 30642 df-mfs 30647 |
This theorem is referenced by: (None) |
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