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 32807 | . 2 ⊢ (𝑇 ∈ mFS → 𝐻:𝑉⟶𝐸) |
5 | eqid 2823 | . . . . . . 7 ⊢ (mType‘𝑇) = (mType‘𝑇) | |
6 | 1, 5, 3 | mvhval 32783 | . . . . . 6 ⊢ (𝑣 ∈ 𝑉 → (𝐻‘𝑣) = 〈((mType‘𝑇)‘𝑣), 〈“𝑣”〉〉) |
7 | 1, 5, 3 | mvhval 32783 | . . . . . 6 ⊢ (𝑤 ∈ 𝑉 → (𝐻‘𝑤) = 〈((mType‘𝑇)‘𝑤), 〈“𝑤”〉〉) |
8 | 6, 7 | eqeqan12d 2840 | . . . . 5 ⊢ ((𝑣 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉) → ((𝐻‘𝑣) = (𝐻‘𝑤) ↔ 〈((mType‘𝑇)‘𝑣), 〈“𝑣”〉〉 = 〈((mType‘𝑇)‘𝑤), 〈“𝑤”〉〉)) |
9 | 8 | adantl 484 | . . . 4 ⊢ ((𝑇 ∈ mFS ∧ (𝑣 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉)) → ((𝐻‘𝑣) = (𝐻‘𝑤) ↔ 〈((mType‘𝑇)‘𝑣), 〈“𝑣”〉〉 = 〈((mType‘𝑇)‘𝑤), 〈“𝑤”〉〉)) |
10 | fvex 6685 | . . . . . . 7 ⊢ ((mType‘𝑇)‘𝑣) ∈ V | |
11 | s1cli 13961 | . . . . . . . 8 ⊢ 〈“𝑣”〉 ∈ Word V | |
12 | 11 | elexi 3515 | . . . . . . 7 ⊢ 〈“𝑣”〉 ∈ V |
13 | 10, 12 | opth 5370 | . . . . . 6 ⊢ (〈((mType‘𝑇)‘𝑣), 〈“𝑣”〉〉 = 〈((mType‘𝑇)‘𝑤), 〈“𝑤”〉〉 ↔ (((mType‘𝑇)‘𝑣) = ((mType‘𝑇)‘𝑤) ∧ 〈“𝑣”〉 = 〈“𝑤”〉)) |
14 | 13 | simprbi 499 | . . . . 5 ⊢ (〈((mType‘𝑇)‘𝑣), 〈“𝑣”〉〉 = 〈((mType‘𝑇)‘𝑤), 〈“𝑤”〉〉 → 〈“𝑣”〉 = 〈“𝑤”〉) |
15 | s111 13971 | . . . . . 6 ⊢ ((𝑣 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉) → (〈“𝑣”〉 = 〈“𝑤”〉 ↔ 𝑣 = 𝑤)) | |
16 | 15 | adantl 484 | . . . . 5 ⊢ ((𝑇 ∈ mFS ∧ (𝑣 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉)) → (〈“𝑣”〉 = 〈“𝑤”〉 ↔ 𝑣 = 𝑤)) |
17 | 14, 16 | syl5ib 246 | . . . 4 ⊢ ((𝑇 ∈ mFS ∧ (𝑣 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉)) → (〈((mType‘𝑇)‘𝑣), 〈“𝑣”〉〉 = 〈((mType‘𝑇)‘𝑤), 〈“𝑤”〉〉 → 𝑣 = 𝑤)) |
18 | 9, 17 | sylbid 242 | . . 3 ⊢ ((𝑇 ∈ mFS ∧ (𝑣 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉)) → ((𝐻‘𝑣) = (𝐻‘𝑤) → 𝑣 = 𝑤)) |
19 | 18 | ralrimivva 3193 | . 2 ⊢ (𝑇 ∈ mFS → ∀𝑣 ∈ 𝑉 ∀𝑤 ∈ 𝑉 ((𝐻‘𝑣) = (𝐻‘𝑤) → 𝑣 = 𝑤)) |
20 | dff13 7015 | . 2 ⊢ (𝐻:𝑉–1-1→𝐸 ↔ (𝐻:𝑉⟶𝐸 ∧ ∀𝑣 ∈ 𝑉 ∀𝑤 ∈ 𝑉 ((𝐻‘𝑣) = (𝐻‘𝑤) → 𝑣 = 𝑤))) | |
21 | 4, 19, 20 | sylanbrc 585 | 1 ⊢ (𝑇 ∈ mFS → 𝐻:𝑉–1-1→𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1537 ∈ wcel 2114 ∀wral 3140 Vcvv 3496 〈cop 4575 ⟶wf 6353 –1-1→wf1 6354 ‘cfv 6357 Word cword 13864 〈“cs1 13951 mVRcmvar 32710 mTypecmty 32711 mExcmex 32716 mVHcmvh 32721 mFScmfs 32725 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-1cn 10597 ax-icn 10598 ax-addcl 10599 ax-addrcl 10600 ax-mulcl 10601 ax-mulrcl 10602 ax-mulcom 10603 ax-addass 10604 ax-mulass 10605 ax-distr 10606 ax-i2m1 10607 ax-1ne0 10608 ax-1rid 10609 ax-rnegex 10610 ax-rrecex 10611 ax-cnre 10612 ax-pre-lttri 10613 ax-pre-lttrn 10614 ax-pre-ltadd 10615 ax-pre-mulgt0 10616 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-riota 7116 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-1o 8104 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-card 9370 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-sub 10874 df-neg 10875 df-nn 11641 df-n0 11901 df-z 11985 df-uz 12247 df-fz 12896 df-fzo 13037 df-hash 13694 df-word 13865 df-s1 13952 df-mrex 32735 df-mex 32736 df-mvh 32741 df-mfs 32745 |
This theorem is referenced by: (None) |
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