Mathbox for Mario Carneiro |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > mvhf1 | Structured version Visualization version GIF version |
Description: The function mapping variables to variable expressions is one-to-one. (Contributed by Mario Carneiro, 18-Jul-2016.) |
Ref | Expression |
---|---|
mvhf.v | ⊢ 𝑉 = (mVR‘𝑇) |
mvhf.e | ⊢ 𝐸 = (mEx‘𝑇) |
mvhf.h | ⊢ 𝐻 = (mVH‘𝑇) |
Ref | Expression |
---|---|
mvhf1 | ⊢ (𝑇 ∈ mFS → 𝐻:𝑉–1-1→𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mvhf.v | . . 3 ⊢ 𝑉 = (mVR‘𝑇) | |
2 | mvhf.e | . . 3 ⊢ 𝐸 = (mEx‘𝑇) | |
3 | mvhf.h | . . 3 ⊢ 𝐻 = (mVH‘𝑇) | |
4 | 1, 2, 3 | mvhf 31455 | . 2 ⊢ (𝑇 ∈ mFS → 𝐻:𝑉⟶𝐸) |
5 | eqid 2622 | . . . . . . 7 ⊢ (mType‘𝑇) = (mType‘𝑇) | |
6 | 1, 5, 3 | mvhval 31431 | . . . . . 6 ⊢ (𝑣 ∈ 𝑉 → (𝐻‘𝑣) = 〈((mType‘𝑇)‘𝑣), 〈“𝑣”〉〉) |
7 | 1, 5, 3 | mvhval 31431 | . . . . . 6 ⊢ (𝑤 ∈ 𝑉 → (𝐻‘𝑤) = 〈((mType‘𝑇)‘𝑤), 〈“𝑤”〉〉) |
8 | 6, 7 | eqeqan12d 2638 | . . . . 5 ⊢ ((𝑣 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉) → ((𝐻‘𝑣) = (𝐻‘𝑤) ↔ 〈((mType‘𝑇)‘𝑣), 〈“𝑣”〉〉 = 〈((mType‘𝑇)‘𝑤), 〈“𝑤”〉〉)) |
9 | 8 | adantl 482 | . . . 4 ⊢ ((𝑇 ∈ mFS ∧ (𝑣 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉)) → ((𝐻‘𝑣) = (𝐻‘𝑤) ↔ 〈((mType‘𝑇)‘𝑣), 〈“𝑣”〉〉 = 〈((mType‘𝑇)‘𝑤), 〈“𝑤”〉〉)) |
10 | fvex 6201 | . . . . . . 7 ⊢ ((mType‘𝑇)‘𝑣) ∈ V | |
11 | s1cli 13384 | . . . . . . . 8 ⊢ 〈“𝑣”〉 ∈ Word V | |
12 | 11 | elexi 3213 | . . . . . . 7 ⊢ 〈“𝑣”〉 ∈ V |
13 | 10, 12 | opth 4945 | . . . . . 6 ⊢ (〈((mType‘𝑇)‘𝑣), 〈“𝑣”〉〉 = 〈((mType‘𝑇)‘𝑤), 〈“𝑤”〉〉 ↔ (((mType‘𝑇)‘𝑣) = ((mType‘𝑇)‘𝑤) ∧ 〈“𝑣”〉 = 〈“𝑤”〉)) |
14 | 13 | simprbi 480 | . . . . 5 ⊢ (〈((mType‘𝑇)‘𝑣), 〈“𝑣”〉〉 = 〈((mType‘𝑇)‘𝑤), 〈“𝑤”〉〉 → 〈“𝑣”〉 = 〈“𝑤”〉) |
15 | s111 13395 | . . . . . 6 ⊢ ((𝑣 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉) → (〈“𝑣”〉 = 〈“𝑤”〉 ↔ 𝑣 = 𝑤)) | |
16 | 15 | adantl 482 | . . . . 5 ⊢ ((𝑇 ∈ mFS ∧ (𝑣 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉)) → (〈“𝑣”〉 = 〈“𝑤”〉 ↔ 𝑣 = 𝑤)) |
17 | 14, 16 | syl5ib 234 | . . . 4 ⊢ ((𝑇 ∈ mFS ∧ (𝑣 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉)) → (〈((mType‘𝑇)‘𝑣), 〈“𝑣”〉〉 = 〈((mType‘𝑇)‘𝑤), 〈“𝑤”〉〉 → 𝑣 = 𝑤)) |
18 | 9, 17 | sylbid 230 | . . 3 ⊢ ((𝑇 ∈ mFS ∧ (𝑣 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉)) → ((𝐻‘𝑣) = (𝐻‘𝑤) → 𝑣 = 𝑤)) |
19 | 18 | ralrimivva 2971 | . 2 ⊢ (𝑇 ∈ mFS → ∀𝑣 ∈ 𝑉 ∀𝑤 ∈ 𝑉 ((𝐻‘𝑣) = (𝐻‘𝑤) → 𝑣 = 𝑤)) |
20 | dff13 6512 | . 2 ⊢ (𝐻:𝑉–1-1→𝐸 ↔ (𝐻:𝑉⟶𝐸 ∧ ∀𝑣 ∈ 𝑉 ∀𝑤 ∈ 𝑉 ((𝐻‘𝑣) = (𝐻‘𝑤) → 𝑣 = 𝑤))) | |
21 | 4, 19, 20 | sylanbrc 698 | 1 ⊢ (𝑇 ∈ mFS → 𝐻:𝑉–1-1→𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ∀wral 2912 Vcvv 3200 〈cop 4183 ⟶wf 5884 –1-1→wf1 5885 ‘cfv 5888 Word cword 13291 〈“cs1 13294 mVRcmvar 31358 mTypecmty 31359 mExcmex 31364 mVHcmvh 31369 mFScmfs 31373 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-int 4476 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-om 7066 df-1st 7168 df-2nd 7169 df-wrecs 7407 df-recs 7468 df-rdg 7506 df-1o 7560 df-er 7742 df-map 7859 df-pm 7860 df-en 7956 df-dom 7957 df-sdom 7958 df-fin 7959 df-card 8765 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269 df-nn 11021 df-n0 11293 df-z 11378 df-uz 11688 df-fz 12327 df-fzo 12466 df-hash 13118 df-word 13299 df-s1 13302 df-mrex 31383 df-mex 31384 df-mvh 31389 df-mfs 31393 |
This theorem is referenced by: (None) |
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