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 30709 | . 2 ⊢ (𝑇 ∈ mFS → 𝐻:𝑉⟶𝐸) |
5 | eqid 2610 | . . . . . . 7 ⊢ (mType‘𝑇) = (mType‘𝑇) | |
6 | 1, 5, 3 | mvhval 30685 | . . . . . 6 ⊢ (𝑣 ∈ 𝑉 → (𝐻‘𝑣) = 〈((mType‘𝑇)‘𝑣), 〈“𝑣”〉〉) |
7 | 1, 5, 3 | mvhval 30685 | . . . . . 6 ⊢ (𝑤 ∈ 𝑉 → (𝐻‘𝑤) = 〈((mType‘𝑇)‘𝑤), 〈“𝑤”〉〉) |
8 | 6, 7 | eqeqan12d 2626 | . . . . 5 ⊢ ((𝑣 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉) → ((𝐻‘𝑣) = (𝐻‘𝑤) ↔ 〈((mType‘𝑇)‘𝑣), 〈“𝑣”〉〉 = 〈((mType‘𝑇)‘𝑤), 〈“𝑤”〉〉)) |
9 | 8 | adantl 481 | . . . 4 ⊢ ((𝑇 ∈ mFS ∧ (𝑣 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉)) → ((𝐻‘𝑣) = (𝐻‘𝑤) ↔ 〈((mType‘𝑇)‘𝑣), 〈“𝑣”〉〉 = 〈((mType‘𝑇)‘𝑤), 〈“𝑤”〉〉)) |
10 | fvex 6113 | . . . . . . 7 ⊢ ((mType‘𝑇)‘𝑣) ∈ V | |
11 | s1cli 13237 | . . . . . . . 8 ⊢ 〈“𝑣”〉 ∈ Word V | |
12 | 11 | elexi 3186 | . . . . . . 7 ⊢ 〈“𝑣”〉 ∈ V |
13 | 10, 12 | opth 4871 | . . . . . 6 ⊢ (〈((mType‘𝑇)‘𝑣), 〈“𝑣”〉〉 = 〈((mType‘𝑇)‘𝑤), 〈“𝑤”〉〉 ↔ (((mType‘𝑇)‘𝑣) = ((mType‘𝑇)‘𝑤) ∧ 〈“𝑣”〉 = 〈“𝑤”〉)) |
14 | 13 | simprbi 479 | . . . . 5 ⊢ (〈((mType‘𝑇)‘𝑣), 〈“𝑣”〉〉 = 〈((mType‘𝑇)‘𝑤), 〈“𝑤”〉〉 → 〈“𝑣”〉 = 〈“𝑤”〉) |
15 | s111 13248 | . . . . . 6 ⊢ ((𝑣 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉) → (〈“𝑣”〉 = 〈“𝑤”〉 ↔ 𝑣 = 𝑤)) | |
16 | 15 | adantl 481 | . . . . 5 ⊢ ((𝑇 ∈ mFS ∧ (𝑣 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉)) → (〈“𝑣”〉 = 〈“𝑤”〉 ↔ 𝑣 = 𝑤)) |
17 | 14, 16 | syl5ib 233 | . . . 4 ⊢ ((𝑇 ∈ mFS ∧ (𝑣 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉)) → (〈((mType‘𝑇)‘𝑣), 〈“𝑣”〉〉 = 〈((mType‘𝑇)‘𝑤), 〈“𝑤”〉〉 → 𝑣 = 𝑤)) |
18 | 9, 17 | sylbid 229 | . . 3 ⊢ ((𝑇 ∈ mFS ∧ (𝑣 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉)) → ((𝐻‘𝑣) = (𝐻‘𝑤) → 𝑣 = 𝑤)) |
19 | 18 | ralrimivva 2954 | . 2 ⊢ (𝑇 ∈ mFS → ∀𝑣 ∈ 𝑉 ∀𝑤 ∈ 𝑉 ((𝐻‘𝑣) = (𝐻‘𝑤) → 𝑣 = 𝑤)) |
20 | dff13 6416 | . 2 ⊢ (𝐻:𝑉–1-1→𝐸 ↔ (𝐻:𝑉⟶𝐸 ∧ ∀𝑣 ∈ 𝑉 ∀𝑤 ∈ 𝑉 ((𝐻‘𝑣) = (𝐻‘𝑤) → 𝑣 = 𝑤))) | |
21 | 4, 19, 20 | sylanbrc 695 | 1 ⊢ (𝑇 ∈ mFS → 𝐻:𝑉–1-1→𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 Vcvv 3173 〈cop 4131 ⟶wf 5800 –1-1→wf1 5801 ‘cfv 5804 Word cword 13146 〈“cs1 13149 mVRcmvar 30612 mTypecmty 30613 mExcmex 30618 mVHcmvh 30623 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-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-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-n0 11170 df-z 11255 df-uz 11564 df-fz 12198 df-fzo 12335 df-hash 12980 df-word 13154 df-s1 13157 df-mrex 30637 df-mex 30638 df-mvh 30643 df-mfs 30647 |
This theorem is referenced by: (None) |
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