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Theorem dibf11N 39627
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 = (LHypβ€˜πΎ)
dibcl.i 𝐼 = ((DIsoBβ€˜πΎ)β€˜π‘Š)
Assertion
Ref Expression
dibf11N ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝐼:dom 𝐼–1-1-ontoβ†’ran 𝐼)

Proof of Theorem dibf11N
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2737 . . . 4 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
2 eqid 2737 . . . 4 (leβ€˜πΎ) = (leβ€˜πΎ)
3 dibcl.h . . . 4 𝐻 = (LHypβ€˜πΎ)
4 dibcl.i . . . 4 𝐼 = ((DIsoBβ€˜πΎ)β€˜π‘Š)
51, 2, 3, 4dibfnN 39622 . . 3 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝐼 Fn {π‘₯ ∈ (Baseβ€˜πΎ) ∣ π‘₯(leβ€˜πΎ)π‘Š})
6 fnfun 6603 . . . 4 (𝐼 Fn {π‘₯ ∈ (Baseβ€˜πΎ) ∣ π‘₯(leβ€˜πΎ)π‘Š} β†’ Fun 𝐼)
7 funfn 6532 . . . 4 (Fun 𝐼 ↔ 𝐼 Fn dom 𝐼)
86, 7sylib 217 . . 3 (𝐼 Fn {π‘₯ ∈ (Baseβ€˜πΎ) ∣ π‘₯(leβ€˜πΎ)π‘Š} β†’ 𝐼 Fn dom 𝐼)
95, 8syl 17 . 2 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝐼 Fn dom 𝐼)
10 eqidd 2738 . 2 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ ran 𝐼 = ran 𝐼)
111, 2, 3, 4dibeldmN 39624 . . . . 5 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (π‘₯ ∈ dom 𝐼 ↔ (π‘₯ ∈ (Baseβ€˜πΎ) ∧ π‘₯(leβ€˜πΎ)π‘Š)))
121, 2, 3, 4dibeldmN 39624 . . . . 5 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (𝑦 ∈ dom 𝐼 ↔ (𝑦 ∈ (Baseβ€˜πΎ) ∧ 𝑦(leβ€˜πΎ)π‘Š)))
1311, 12anbi12d 632 . . . 4 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ ((π‘₯ ∈ dom 𝐼 ∧ 𝑦 ∈ dom 𝐼) ↔ ((π‘₯ ∈ (Baseβ€˜πΎ) ∧ π‘₯(leβ€˜πΎ)π‘Š) ∧ (𝑦 ∈ (Baseβ€˜πΎ) ∧ 𝑦(leβ€˜πΎ)π‘Š))))
141, 2, 3, 4dib11N 39626 . . . . . 6 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘₯ ∈ (Baseβ€˜πΎ) ∧ π‘₯(leβ€˜πΎ)π‘Š) ∧ (𝑦 ∈ (Baseβ€˜πΎ) ∧ 𝑦(leβ€˜πΎ)π‘Š)) β†’ ((πΌβ€˜π‘₯) = (πΌβ€˜π‘¦) ↔ π‘₯ = 𝑦))
1514biimpd 228 . . . . 5 (((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) ∧ (π‘₯ ∈ (Baseβ€˜πΎ) ∧ π‘₯(leβ€˜πΎ)π‘Š) ∧ (𝑦 ∈ (Baseβ€˜πΎ) ∧ 𝑦(leβ€˜πΎ)π‘Š)) β†’ ((πΌβ€˜π‘₯) = (πΌβ€˜π‘¦) β†’ π‘₯ = 𝑦))
16153expib 1123 . . . 4 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (((π‘₯ ∈ (Baseβ€˜πΎ) ∧ π‘₯(leβ€˜πΎ)π‘Š) ∧ (𝑦 ∈ (Baseβ€˜πΎ) ∧ 𝑦(leβ€˜πΎ)π‘Š)) β†’ ((πΌβ€˜π‘₯) = (πΌβ€˜π‘¦) β†’ π‘₯ = 𝑦)))
1713, 16sylbid 239 . . 3 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ ((π‘₯ ∈ dom 𝐼 ∧ 𝑦 ∈ dom 𝐼) β†’ ((πΌβ€˜π‘₯) = (πΌβ€˜π‘¦) β†’ π‘₯ = 𝑦)))
1817ralrimivv 3196 . 2 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ βˆ€π‘₯ ∈ dom πΌβˆ€π‘¦ ∈ dom 𝐼((πΌβ€˜π‘₯) = (πΌβ€˜π‘¦) β†’ π‘₯ = 𝑦))
19 dff1o6 7222 . 2 (𝐼:dom 𝐼–1-1-ontoβ†’ran 𝐼 ↔ (𝐼 Fn dom 𝐼 ∧ ran 𝐼 = ran 𝐼 ∧ βˆ€π‘₯ ∈ dom πΌβˆ€π‘¦ ∈ dom 𝐼((πΌβ€˜π‘₯) = (πΌβ€˜π‘¦) β†’ π‘₯ = 𝑦)))
209, 10, 18, 19syl3anbrc 1344 1 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝐼:dom 𝐼–1-1-ontoβ†’ran 𝐼)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3065  {crab 3408   class class class wbr 5106  dom cdm 5634  ran crn 5635  Fun wfun 6491   Fn wfn 6492  β€“1-1-ontoβ†’wf1o 6496  β€˜cfv 6497  Basecbs 17084  lecple 17141  HLchlt 37815  LHypclh 38450  DIsoBcdib 39604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-riotaBAD 37418
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3354  df-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-iin 4958  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923  df-undef 8205  df-map 8768  df-proset 18185  df-poset 18203  df-plt 18220  df-lub 18236  df-glb 18237  df-join 18238  df-meet 18239  df-p0 18315  df-p1 18316  df-lat 18322  df-clat 18389  df-oposet 37641  df-ol 37643  df-oml 37644  df-covers 37731  df-ats 37732  df-atl 37763  df-cvlat 37787  df-hlat 37816  df-llines 37964  df-lplanes 37965  df-lvols 37966  df-lines 37967  df-psubsp 37969  df-pmap 37970  df-padd 38262  df-lhyp 38454  df-laut 38455  df-ldil 38570  df-ltrn 38571  df-trl 38625  df-disoa 39495  df-dib 39605
This theorem is referenced by:  dibintclN  39633
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