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Theorem dibf11N 40666
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 2728 . . . 4 (Baseβ€˜πΎ) = (Baseβ€˜πΎ)
2 eqid 2728 . . . 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 40661 . . 3 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ 𝐼 Fn {π‘₯ ∈ (Baseβ€˜πΎ) ∣ π‘₯(leβ€˜πΎ)π‘Š})
6 fnfun 6659 . . . 4 (𝐼 Fn {π‘₯ ∈ (Baseβ€˜πΎ) ∣ π‘₯(leβ€˜πΎ)π‘Š} β†’ Fun 𝐼)
7 funfn 6588 . . . 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 2729 . 2 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ ran 𝐼 = ran 𝐼)
111, 2, 3, 4dibeldmN 40663 . . . . 5 ((𝐾 ∈ HL ∧ π‘Š ∈ 𝐻) β†’ (π‘₯ ∈ dom 𝐼 ↔ (π‘₯ ∈ (Baseβ€˜πΎ) ∧ π‘₯(leβ€˜πΎ)π‘Š)))
121, 2, 3, 4dibeldmN 40663 . . . . 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 40665 . . . . . 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 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 7290 . 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 394   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  βˆ€wral 3058  {crab 3430   class class class wbr 5152  dom cdm 5682  ran crn 5683  Fun wfun 6547   Fn wfn 6548  β€“1-1-ontoβ†’wf1o 6552  β€˜cfv 6553  Basecbs 17187  lecple 17247  HLchlt 38854  LHypclh 39489  DIsoBcdib 40643
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 2166  ax-ext 2699  ax-rep 5289  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746  ax-riotaBAD 38457
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-iin 5003  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-f1 6558  df-fo 6559  df-f1o 6560  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-1st 7999  df-2nd 8000  df-undef 8285  df-map 8853  df-proset 18294  df-poset 18312  df-plt 18329  df-lub 18345  df-glb 18346  df-join 18347  df-meet 18348  df-p0 18424  df-p1 18425  df-lat 18431  df-clat 18498  df-oposet 38680  df-ol 38682  df-oml 38683  df-covers 38770  df-ats 38771  df-atl 38802  df-cvlat 38826  df-hlat 38855  df-llines 39003  df-lplanes 39004  df-lvols 39005  df-lines 39006  df-psubsp 39008  df-pmap 39009  df-padd 39301  df-lhyp 39493  df-laut 39494  df-ldil 39609  df-ltrn 39610  df-trl 39664  df-disoa 40534  df-dib 40644
This theorem is referenced by:  dibintclN  40672
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