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Theorem dibf11N 40544
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 2726 . . . 4 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
2 eqid 2726 . . . 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 40539 . . 3 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝐼 Fn {π‘₯ ∈ (Baseβ€˜πΎ) ∣ π‘₯(leβ€˜πΎ)π‘Š})
6 fnfun 6642 . . . 4 (𝐼 Fn {π‘₯ ∈ (Baseβ€˜πΎ) ∣ π‘₯(leβ€˜πΎ)π‘Š} β†’ Fun 𝐼)
7 funfn 6571 . . . 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 2727 . 2 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ ran 𝐼 = ran 𝐼)
111, 2, 3, 4dibeldmN 40541 . . . . 5 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (π‘₯ ∈ dom 𝐼 ↔ (π‘₯ ∈ (Baseβ€˜πΎ) ∧ π‘₯(leβ€˜πΎ)π‘Š)))
121, 2, 3, 4dibeldmN 40541 . . . . 5 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (𝑦 ∈ dom 𝐼 ↔ (𝑦 ∈ (Baseβ€˜πΎ) ∧ 𝑦(leβ€˜πΎ)π‘Š)))
1311, 12anbi12d 630 . . . 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 40543 . . . . . 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 1119 . . . 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 3192 . 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 7268 . 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 1340 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 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055  {crab 3426   class class class wbr 5141  dom cdm 5669  ran crn 5670  Fun wfun 6530   Fn wfn 6531  β€“1-1-ontoβ†’wf1o 6535  β€˜cfv 6536  Basecbs 17150  lecple 17210  HLchlt 38732  LHypclh 39367  DIsoBcdib 40521
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721  ax-riotaBAD 38335
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-iin 4993  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7360  df-ov 7407  df-oprab 7408  df-mpo 7409  df-1st 7971  df-2nd 7972  df-undef 8256  df-map 8821  df-proset 18257  df-poset 18275  df-plt 18292  df-lub 18308  df-glb 18309  df-join 18310  df-meet 18311  df-p0 18387  df-p1 18388  df-lat 18394  df-clat 18461  df-oposet 38558  df-ol 38560  df-oml 38561  df-covers 38648  df-ats 38649  df-atl 38680  df-cvlat 38704  df-hlat 38733  df-llines 38881  df-lplanes 38882  df-lvols 38883  df-lines 38884  df-psubsp 38886  df-pmap 38887  df-padd 39179  df-lhyp 39371  df-laut 39372  df-ldil 39487  df-ltrn 39488  df-trl 39542  df-disoa 40412  df-dib 40522
This theorem is referenced by:  dibintclN  40550
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