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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 35563 | . 2 ⊢ (𝑇 ∈ mFS → 𝐻:𝑉⟶𝐸) |
| 5 | eqid 2737 | . . . . . . 7 ⊢ (mType‘𝑇) = (mType‘𝑇) | |
| 6 | 1, 5, 3 | mvhval 35539 | . . . . . 6 ⊢ (𝑣 ∈ 𝑉 → (𝐻‘𝑣) = 〈((mType‘𝑇)‘𝑣), 〈“𝑣”〉〉) |
| 7 | 1, 5, 3 | mvhval 35539 | . . . . . 6 ⊢ (𝑤 ∈ 𝑉 → (𝐻‘𝑤) = 〈((mType‘𝑇)‘𝑤), 〈“𝑤”〉〉) |
| 8 | 6, 7 | eqeqan12d 2751 | . . . . 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 6919 | . . . . . . 7 ⊢ ((mType‘𝑇)‘𝑣) ∈ V | |
| 11 | s1cli 14643 | . . . . . . . 8 ⊢ 〈“𝑣”〉 ∈ Word V | |
| 12 | 11 | elexi 3503 | . . . . . . 7 ⊢ 〈“𝑣”〉 ∈ V |
| 13 | 10, 12 | opth 5481 | . . . . . 6 ⊢ (〈((mType‘𝑇)‘𝑣), 〈“𝑣”〉〉 = 〈((mType‘𝑇)‘𝑤), 〈“𝑤”〉〉 ↔ (((mType‘𝑇)‘𝑣) = ((mType‘𝑇)‘𝑤) ∧ 〈“𝑣”〉 = 〈“𝑤”〉)) |
| 14 | 13 | simprbi 496 | . . . . 5 ⊢ (〈((mType‘𝑇)‘𝑣), 〈“𝑣”〉〉 = 〈((mType‘𝑇)‘𝑤), 〈“𝑤”〉〉 → 〈“𝑣”〉 = 〈“𝑤”〉) |
| 15 | s111 14653 | . . . . . 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 | imbitrid 244 | . . . 4 ⊢ ((𝑇 ∈ mFS ∧ (𝑣 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉)) → (〈((mType‘𝑇)‘𝑣), 〈“𝑣”〉〉 = 〈((mType‘𝑇)‘𝑤), 〈“𝑤”〉〉 → 𝑣 = 𝑤)) |
| 18 | 9, 17 | sylbid 240 | . . 3 ⊢ ((𝑇 ∈ mFS ∧ (𝑣 ∈ 𝑉 ∧ 𝑤 ∈ 𝑉)) → ((𝐻‘𝑣) = (𝐻‘𝑤) → 𝑣 = 𝑤)) |
| 19 | 18 | ralrimivva 3202 | . 2 ⊢ (𝑇 ∈ mFS → ∀𝑣 ∈ 𝑉 ∀𝑤 ∈ 𝑉 ((𝐻‘𝑣) = (𝐻‘𝑤) → 𝑣 = 𝑤)) |
| 20 | dff13 7275 | . 2 ⊢ (𝐻:𝑉–1-1→𝐸 ↔ (𝐻:𝑉⟶𝐸 ∧ ∀𝑣 ∈ 𝑉 ∀𝑤 ∈ 𝑉 ((𝐻‘𝑣) = (𝐻‘𝑤) → 𝑣 = 𝑤))) | |
| 21 | 4, 19, 20 | sylanbrc 583 | 1 ⊢ (𝑇 ∈ mFS → 𝐻:𝑉–1-1→𝐸) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 Vcvv 3480 〈cop 4632 ⟶wf 6557 –1-1→wf1 6558 ‘cfv 6561 Word cword 14552 〈“cs1 14633 mVRcmvar 35466 mTypecmty 35467 mExcmex 35472 mVHcmvh 35477 mFScmfs 35481 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-1cn 11213 ax-icn 11214 ax-addcl 11215 ax-addrcl 11216 ax-mulcl 11217 ax-mulrcl 11218 ax-mulcom 11219 ax-addass 11220 ax-mulass 11221 ax-distr 11222 ax-i2m1 11223 ax-1ne0 11224 ax-1rid 11225 ax-rnegex 11226 ax-rrecex 11227 ax-cnre 11228 ax-pre-lttri 11229 ax-pre-lttrn 11230 ax-pre-ltadd 11231 ax-pre-mulgt0 11232 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-fin 8989 df-card 9979 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-sub 11494 df-neg 11495 df-nn 12267 df-n0 12527 df-z 12614 df-uz 12879 df-fz 13548 df-fzo 13695 df-hash 14370 df-word 14553 df-s1 14634 df-mrex 35491 df-mex 35492 df-mvh 35497 df-mfs 35501 |
| This theorem is referenced by: (None) |
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