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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 32001 | . 2 ⊢ (𝑇 ∈ mFS → 𝐻:𝑉⟶𝐸) |
5 | eqid 2825 | . . . . . . 7 ⊢ (mType‘𝑇) = (mType‘𝑇) | |
6 | 1, 5, 3 | mvhval 31977 | . . . . . 6 ⊢ (𝑣 ∈ 𝑉 → (𝐻‘𝑣) = 〈((mType‘𝑇)‘𝑣), 〈“𝑣”〉〉) |
7 | 1, 5, 3 | mvhval 31977 | . . . . . 6 ⊢ (𝑤 ∈ 𝑉 → (𝐻‘𝑤) = 〈((mType‘𝑇)‘𝑤), 〈“𝑤”〉〉) |
8 | 6, 7 | eqeqan12d 2841 | . . . . 5 ⊢ ((𝑣 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉) → ((𝐻‘𝑣) = (𝐻‘𝑤) ↔ 〈((mType‘𝑇)‘𝑣), 〈“𝑣”〉〉 = 〈((mType‘𝑇)‘𝑤), 〈“𝑤”〉〉)) |
9 | 8 | adantl 475 | . . . 4 ⊢ ((𝑇 ∈ mFS ∧ (𝑣 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉)) → ((𝐻‘𝑣) = (𝐻‘𝑤) ↔ 〈((mType‘𝑇)‘𝑣), 〈“𝑣”〉〉 = 〈((mType‘𝑇)‘𝑤), 〈“𝑤”〉〉)) |
10 | fvex 6446 | . . . . . . 7 ⊢ ((mType‘𝑇)‘𝑣) ∈ V | |
11 | s1cli 13665 | . . . . . . . 8 ⊢ 〈“𝑣”〉 ∈ Word V | |
12 | 11 | elexi 3430 | . . . . . . 7 ⊢ 〈“𝑣”〉 ∈ V |
13 | 10, 12 | opth 5165 | . . . . . 6 ⊢ (〈((mType‘𝑇)‘𝑣), 〈“𝑣”〉〉 = 〈((mType‘𝑇)‘𝑤), 〈“𝑤”〉〉 ↔ (((mType‘𝑇)‘𝑣) = ((mType‘𝑇)‘𝑤) ∧ 〈“𝑣”〉 = 〈“𝑤”〉)) |
14 | 13 | simprbi 492 | . . . . 5 ⊢ (〈((mType‘𝑇)‘𝑣), 〈“𝑣”〉〉 = 〈((mType‘𝑇)‘𝑤), 〈“𝑤”〉〉 → 〈“𝑣”〉 = 〈“𝑤”〉) |
15 | s111 13675 | . . . . . 6 ⊢ ((𝑣 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉) → (〈“𝑣”〉 = 〈“𝑤”〉 ↔ 𝑣 = 𝑤)) | |
16 | 15 | adantl 475 | . . . . 5 ⊢ ((𝑇 ∈ mFS ∧ (𝑣 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉)) → (〈“𝑣”〉 = 〈“𝑤”〉 ↔ 𝑣 = 𝑤)) |
17 | 14, 16 | syl5ib 236 | . . . 4 ⊢ ((𝑇 ∈ mFS ∧ (𝑣 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉)) → (〈((mType‘𝑇)‘𝑣), 〈“𝑣”〉〉 = 〈((mType‘𝑇)‘𝑤), 〈“𝑤”〉〉 → 𝑣 = 𝑤)) |
18 | 9, 17 | sylbid 232 | . . 3 ⊢ ((𝑇 ∈ mFS ∧ (𝑣 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉)) → ((𝐻‘𝑣) = (𝐻‘𝑤) → 𝑣 = 𝑤)) |
19 | 18 | ralrimivva 3180 | . 2 ⊢ (𝑇 ∈ mFS → ∀𝑣 ∈ 𝑉 ∀𝑤 ∈ 𝑉 ((𝐻‘𝑣) = (𝐻‘𝑤) → 𝑣 = 𝑤)) |
20 | dff13 6767 | . 2 ⊢ (𝐻:𝑉–1-1→𝐸 ↔ (𝐻:𝑉⟶𝐸 ∧ ∀𝑣 ∈ 𝑉 ∀𝑤 ∈ 𝑉 ((𝐻‘𝑣) = (𝐻‘𝑤) → 𝑣 = 𝑤))) | |
21 | 4, 19, 20 | sylanbrc 580 | 1 ⊢ (𝑇 ∈ mFS → 𝐻:𝑉–1-1→𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ∀wral 3117 Vcvv 3414 〈cop 4403 ⟶wf 6119 –1-1→wf1 6120 ‘cfv 6123 Word cword 13574 〈“cs1 13655 mVRcmvar 31904 mTypecmty 31905 mExcmex 31910 mVHcmvh 31915 mFScmfs 31919 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 ax-cnex 10308 ax-resscn 10309 ax-1cn 10310 ax-icn 10311 ax-addcl 10312 ax-addrcl 10313 ax-mulcl 10314 ax-mulrcl 10315 ax-mulcom 10316 ax-addass 10317 ax-mulass 10318 ax-distr 10319 ax-i2m1 10320 ax-1ne0 10321 ax-1rid 10322 ax-rnegex 10323 ax-rrecex 10324 ax-cnre 10325 ax-pre-lttri 10326 ax-pre-lttrn 10327 ax-pre-ltadd 10328 ax-pre-mulgt0 10329 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-nel 3103 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-int 4698 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-riota 6866 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-1st 7428 df-2nd 7429 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-1o 7826 df-er 8009 df-map 8124 df-pm 8125 df-en 8223 df-dom 8224 df-sdom 8225 df-fin 8226 df-card 9078 df-pnf 10393 df-mnf 10394 df-xr 10395 df-ltxr 10396 df-le 10397 df-sub 10587 df-neg 10588 df-nn 11351 df-n0 11619 df-z 11705 df-uz 11969 df-fz 12620 df-fzo 12761 df-hash 13411 df-word 13575 df-s1 13656 df-mrex 31929 df-mex 31930 df-mvh 31935 df-mfs 31939 |
This theorem is referenced by: (None) |
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