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Theorem dibf11N 38289
Description: The partial isomorphism A for a lattice 𝐾 is a one-to-one function. Part of Lemma M of [Crawley] p. 120 line 27. (Contributed by NM, 4-Dec-2013.) (New usage is discouraged.)
Hypotheses
Ref Expression
dibcl.h 𝐻 = (LHyp‘𝐾)
dibcl.i 𝐼 = ((DIsoB‘𝐾)‘𝑊)
Assertion
Ref Expression
dibf11N ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐼:dom 𝐼1-1-onto→ran 𝐼)

Proof of Theorem dibf11N
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2819 . . . 4 (Base‘𝐾) = (Base‘𝐾)
2 eqid 2819 . . . 4 (le‘𝐾) = (le‘𝐾)
3 dibcl.h . . . 4 𝐻 = (LHyp‘𝐾)
4 dibcl.i . . . 4 𝐼 = ((DIsoB‘𝐾)‘𝑊)
51, 2, 3, 4dibfnN 38284 . . 3 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐼 Fn {𝑥 ∈ (Base‘𝐾) ∣ 𝑥(le‘𝐾)𝑊})
6 fnfun 6446 . . . 4 (𝐼 Fn {𝑥 ∈ (Base‘𝐾) ∣ 𝑥(le‘𝐾)𝑊} → Fun 𝐼)
7 funfn 6378 . . . 4 (Fun 𝐼𝐼 Fn dom 𝐼)
86, 7sylib 220 . . 3 (𝐼 Fn {𝑥 ∈ (Base‘𝐾) ∣ 𝑥(le‘𝐾)𝑊} → 𝐼 Fn dom 𝐼)
95, 8syl 17 . 2 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐼 Fn dom 𝐼)
10 eqidd 2820 . 2 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ran 𝐼 = ran 𝐼)
111, 2, 3, 4dibeldmN 38286 . . . . 5 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (𝑥 ∈ dom 𝐼 ↔ (𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊)))
121, 2, 3, 4dibeldmN 38286 . . . . 5 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (𝑦 ∈ dom 𝐼 ↔ (𝑦 ∈ (Base‘𝐾) ∧ 𝑦(le‘𝐾)𝑊)))
1311, 12anbi12d 632 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ((𝑥 ∈ dom 𝐼𝑦 ∈ dom 𝐼) ↔ ((𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊) ∧ (𝑦 ∈ (Base‘𝐾) ∧ 𝑦(le‘𝐾)𝑊))))
141, 2, 3, 4dib11N 38288 . . . . . 6 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊) ∧ (𝑦 ∈ (Base‘𝐾) ∧ 𝑦(le‘𝐾)𝑊)) → ((𝐼𝑥) = (𝐼𝑦) ↔ 𝑥 = 𝑦))
1514biimpd 231 . . . . 5 (((𝐾 ∈ HL ∧ 𝑊𝐻) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊) ∧ (𝑦 ∈ (Base‘𝐾) ∧ 𝑦(le‘𝐾)𝑊)) → ((𝐼𝑥) = (𝐼𝑦) → 𝑥 = 𝑦))
16153expib 1116 . . . 4 ((𝐾 ∈ HL ∧ 𝑊𝐻) → (((𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊) ∧ (𝑦 ∈ (Base‘𝐾) ∧ 𝑦(le‘𝐾)𝑊)) → ((𝐼𝑥) = (𝐼𝑦) → 𝑥 = 𝑦)))
1713, 16sylbid 242 . . 3 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ((𝑥 ∈ dom 𝐼𝑦 ∈ dom 𝐼) → ((𝐼𝑥) = (𝐼𝑦) → 𝑥 = 𝑦)))
1817ralrimivv 3188 . 2 ((𝐾 ∈ HL ∧ 𝑊𝐻) → ∀𝑥 ∈ dom 𝐼𝑦 ∈ dom 𝐼((𝐼𝑥) = (𝐼𝑦) → 𝑥 = 𝑦))
19 dff1o6 7024 . 2 (𝐼:dom 𝐼1-1-onto→ran 𝐼 ↔ (𝐼 Fn dom 𝐼 ∧ ran 𝐼 = ran 𝐼 ∧ ∀𝑥 ∈ dom 𝐼𝑦 ∈ dom 𝐼((𝐼𝑥) = (𝐼𝑦) → 𝑥 = 𝑦)))
209, 10, 18, 19syl3anbrc 1337 1 ((𝐾 ∈ HL ∧ 𝑊𝐻) → 𝐼:dom 𝐼1-1-onto→ran 𝐼)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  w3a 1081   = wceq 1530  wcel 2107  wral 3136  {crab 3140   class class class wbr 5057  dom cdm 5548  ran crn 5549  Fun wfun 6342   Fn wfn 6343  1-1-ontowf1o 6347  cfv 6348  Basecbs 16475  lecple 16564  HLchlt 36478  LHypclh 37112  DIsoBcdib 38266
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453  ax-riotaBAD 36081
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rmo 3144  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-riota 7106  df-ov 7151  df-oprab 7152  df-mpo 7153  df-1st 7681  df-2nd 7682  df-undef 7931  df-map 8400  df-proset 17530  df-poset 17548  df-plt 17560  df-lub 17576  df-glb 17577  df-join 17578  df-meet 17579  df-p0 17641  df-p1 17642  df-lat 17648  df-clat 17710  df-oposet 36304  df-ol 36306  df-oml 36307  df-covers 36394  df-ats 36395  df-atl 36426  df-cvlat 36450  df-hlat 36479  df-llines 36626  df-lplanes 36627  df-lvols 36628  df-lines 36629  df-psubsp 36631  df-pmap 36632  df-padd 36924  df-lhyp 37116  df-laut 37117  df-ldil 37232  df-ltrn 37233  df-trl 37287  df-disoa 38157  df-dib 38267
This theorem is referenced by:  dibintclN  38295
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