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Mirrors > Home > MPE Home > Th. List > Mathboxes > mapdcnv11N | Structured version Visualization version GIF version |
Description: The converse of the map defined by df-mapd 39893 is one-to-one. (Contributed by NM, 13-Mar-2015.) (New usage is discouraged.) |
Ref | Expression |
---|---|
mapdcnvord.h | ⊢ 𝐻 = (LHyp‘𝐾) |
mapdcnvord.m | ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) |
mapdcnvord.k | ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) |
mapdcnvord.x | ⊢ (𝜑 → 𝑋 ∈ ran 𝑀) |
mapdcnvord.y | ⊢ (𝜑 → 𝑌 ∈ ran 𝑀) |
Ref | Expression |
---|---|
mapdcnv11N | ⊢ (𝜑 → ((◡𝑀‘𝑋) = (◡𝑀‘𝑌) ↔ 𝑋 = 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mapdcnvord.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
2 | mapdcnvord.m | . . . 4 ⊢ 𝑀 = ((mapd‘𝐾)‘𝑊) | |
3 | mapdcnvord.k | . . . 4 ⊢ (𝜑 → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
4 | mapdcnvord.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ ran 𝑀) | |
5 | mapdcnvord.y | . . . 4 ⊢ (𝜑 → 𝑌 ∈ ran 𝑀) | |
6 | 1, 2, 3, 4, 5 | mapdcnvordN 39926 | . . 3 ⊢ (𝜑 → ((◡𝑀‘𝑋) ⊆ (◡𝑀‘𝑌) ↔ 𝑋 ⊆ 𝑌)) |
7 | 1, 2, 3, 5, 4 | mapdcnvordN 39926 | . . 3 ⊢ (𝜑 → ((◡𝑀‘𝑌) ⊆ (◡𝑀‘𝑋) ↔ 𝑌 ⊆ 𝑋)) |
8 | 6, 7 | anbi12d 631 | . 2 ⊢ (𝜑 → (((◡𝑀‘𝑋) ⊆ (◡𝑀‘𝑌) ∧ (◡𝑀‘𝑌) ⊆ (◡𝑀‘𝑋)) ↔ (𝑋 ⊆ 𝑌 ∧ 𝑌 ⊆ 𝑋))) |
9 | eqss 3947 | . 2 ⊢ ((◡𝑀‘𝑋) = (◡𝑀‘𝑌) ↔ ((◡𝑀‘𝑋) ⊆ (◡𝑀‘𝑌) ∧ (◡𝑀‘𝑌) ⊆ (◡𝑀‘𝑋))) | |
10 | eqss 3947 | . 2 ⊢ (𝑋 = 𝑌 ↔ (𝑋 ⊆ 𝑌 ∧ 𝑌 ⊆ 𝑋)) | |
11 | 8, 9, 10 | 3bitr4g 313 | 1 ⊢ (𝜑 → ((◡𝑀‘𝑋) = (◡𝑀‘𝑌) ↔ 𝑋 = 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1540 ∈ wcel 2105 ⊆ wss 3898 ◡ccnv 5619 ran crn 5621 ‘cfv 6479 HLchlt 37617 LHypclh 38252 mapdcmpd 39892 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2707 ax-rep 5229 ax-sep 5243 ax-nul 5250 ax-pow 5308 ax-pr 5372 ax-un 7650 ax-cnex 11028 ax-resscn 11029 ax-1cn 11030 ax-icn 11031 ax-addcl 11032 ax-addrcl 11033 ax-mulcl 11034 ax-mulrcl 11035 ax-mulcom 11036 ax-addass 11037 ax-mulass 11038 ax-distr 11039 ax-i2m1 11040 ax-1ne0 11041 ax-1rid 11042 ax-rnegex 11043 ax-rrecex 11044 ax-cnre 11045 ax-pre-lttri 11046 ax-pre-lttrn 11047 ax-pre-ltadd 11048 ax-pre-mulgt0 11049 ax-riotaBAD 37220 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3349 df-reu 3350 df-rab 3404 df-v 3443 df-sbc 3728 df-csb 3844 df-dif 3901 df-un 3903 df-in 3905 df-ss 3915 df-pss 3917 df-nul 4270 df-if 4474 df-pw 4549 df-sn 4574 df-pr 4576 df-tp 4578 df-op 4580 df-uni 4853 df-int 4895 df-iun 4943 df-iin 4944 df-br 5093 df-opab 5155 df-mpt 5176 df-tr 5210 df-id 5518 df-eprel 5524 df-po 5532 df-so 5533 df-fr 5575 df-we 5577 df-xp 5626 df-rel 5627 df-cnv 5628 df-co 5629 df-dm 5630 df-rn 5631 df-res 5632 df-ima 5633 df-pred 6238 df-ord 6305 df-on 6306 df-lim 6307 df-suc 6308 df-iota 6431 df-fun 6481 df-fn 6482 df-f 6483 df-f1 6484 df-fo 6485 df-f1o 6486 df-fv 6487 df-riota 7293 df-ov 7340 df-oprab 7341 df-mpo 7342 df-of 7595 df-om 7781 df-1st 7899 df-2nd 7900 df-tpos 8112 df-undef 8159 df-frecs 8167 df-wrecs 8198 df-recs 8272 df-rdg 8311 df-1o 8367 df-er 8569 df-map 8688 df-en 8805 df-dom 8806 df-sdom 8807 df-fin 8808 df-pnf 11112 df-mnf 11113 df-xr 11114 df-ltxr 11115 df-le 11116 df-sub 11308 df-neg 11309 df-nn 12075 df-2 12137 df-3 12138 df-4 12139 df-5 12140 df-6 12141 df-n0 12335 df-z 12421 df-uz 12684 df-fz 13341 df-struct 16945 df-sets 16962 df-slot 16980 df-ndx 16992 df-base 17010 df-ress 17039 df-plusg 17072 df-mulr 17073 df-sca 17075 df-vsca 17076 df-0g 17249 df-mre 17392 df-mrc 17393 df-acs 17395 df-proset 18110 df-poset 18128 df-plt 18145 df-lub 18161 df-glb 18162 df-join 18163 df-meet 18164 df-p0 18240 df-p1 18241 df-lat 18247 df-clat 18314 df-mgm 18423 df-sgrp 18472 df-mnd 18483 df-submnd 18528 df-grp 18676 df-minusg 18677 df-sbg 18678 df-subg 18848 df-cntz 19019 df-oppg 19046 df-lsm 19337 df-cmn 19483 df-abl 19484 df-mgp 19816 df-ur 19833 df-ring 19880 df-oppr 19957 df-dvdsr 19978 df-unit 19979 df-invr 20009 df-dvr 20020 df-drng 20095 df-lmod 20231 df-lss 20300 df-lsp 20340 df-lvec 20471 df-lsatoms 37243 df-lshyp 37244 df-lcv 37286 df-lfl 37325 df-lkr 37353 df-ldual 37391 df-oposet 37443 df-ol 37445 df-oml 37446 df-covers 37533 df-ats 37534 df-atl 37565 df-cvlat 37589 df-hlat 37618 df-llines 37766 df-lplanes 37767 df-lvols 37768 df-lines 37769 df-psubsp 37771 df-pmap 37772 df-padd 38064 df-lhyp 38256 df-laut 38257 df-ldil 38372 df-ltrn 38373 df-trl 38427 df-tgrp 39011 df-tendo 39023 df-edring 39025 df-dveca 39271 df-disoa 39297 df-dvech 39347 df-dib 39407 df-dic 39441 df-dih 39497 df-doch 39616 df-djh 39663 df-mapd 39893 |
This theorem is referenced by: hdmaprnlem3uN 40119 hdmaprnlem9N 40125 hdmaprnlem3eN 40126 |
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