Theorem diaf11N 41043

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
dia1o.h	⊢ 𝐻 = (LHyp‘𝐾)
dia1o.i	⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)

Assertion

Ref	Expression
diaf11N	⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐼:dom 𝐼–1-1-onto→ran 𝐼)

Proof of Theorem diaf11N

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2729	. . . 4 ⊢ (Base‘𝐾) = (Base‘𝐾)
2		eqid 2729	. . . 4 ⊢ (le‘𝐾) = (le‘𝐾)
3		dia1o.h	. . . 4 ⊢ 𝐻 = (LHyp‘𝐾)
4		dia1o.i	. . . 4 ⊢ 𝐼 = ((DIsoA‘𝐾)‘𝑊)
5	1, 2, 3, 4	diafn 41028	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐼 Fn {𝑥 ∈ (Base‘𝐾) ∣ 𝑥(le‘𝐾)𝑊})
6		fnfun 6618	. . . 4 ⊢ (𝐼 Fn {𝑥 ∈ (Base‘𝐾) ∣ 𝑥(le‘𝐾)𝑊} → Fun 𝐼)
7		funfn 6546	. . . 4 ⊢ (Fun 𝐼 ↔ 𝐼 Fn dom 𝐼)
8	6, 7	sylib 218	. . 3 ⊢ (𝐼 Fn {𝑥 ∈ (Base‘𝐾) ∣ 𝑥(le‘𝐾)𝑊} → 𝐼 Fn dom 𝐼)
9	5, 8	syl 17	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐼 Fn dom 𝐼)
10		eqidd 2730	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ran 𝐼 = ran 𝐼)
11	1, 2, 3, 4	diaeldm 41030	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑥 ∈ dom 𝐼 ↔ (𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊)))
12	1, 2, 3, 4	diaeldm 41030	. . . . 5 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (𝑦 ∈ dom 𝐼 ↔ (𝑦 ∈ (Base‘𝐾) ∧ 𝑦(le‘𝐾)𝑊)))
13	11, 12	anbi12d 632	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ((𝑥 ∈ dom 𝐼 ∧ 𝑦 ∈ dom 𝐼) ↔ ((𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊) ∧ (𝑦 ∈ (Base‘𝐾) ∧ 𝑦(le‘𝐾)𝑊))))
14	1, 2, 3, 4	dia11N 41042	. . . . . 6 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊) ∧ (𝑦 ∈ (Base‘𝐾) ∧ 𝑦(le‘𝐾)𝑊)) → ((𝐼‘𝑥) = (𝐼‘𝑦) ↔ 𝑥 = 𝑦))
15	14	biimpd 229	. . . . 5 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊) ∧ (𝑦 ∈ (Base‘𝐾) ∧ 𝑦(le‘𝐾)𝑊)) → ((𝐼‘𝑥) = (𝐼‘𝑦) → 𝑥 = 𝑦))
16	15	3expib 1122	. . . 4 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → (((𝑥 ∈ (Base‘𝐾) ∧ 𝑥(le‘𝐾)𝑊) ∧ (𝑦 ∈ (Base‘𝐾) ∧ 𝑦(le‘𝐾)𝑊)) → ((𝐼‘𝑥) = (𝐼‘𝑦) → 𝑥 = 𝑦)))
17	13, 16	sylbid 240	. . 3 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ((𝑥 ∈ dom 𝐼 ∧ 𝑦 ∈ dom 𝐼) → ((𝐼‘𝑥) = (𝐼‘𝑦) → 𝑥 = 𝑦)))
18	17	ralrimivv 3178	. 2 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → ∀𝑥 ∈ dom 𝐼∀𝑦 ∈ dom 𝐼((𝐼‘𝑥) = (𝐼‘𝑦) → 𝑥 = 𝑦))
19		dff1o6 7250	. 2 ⊢ (𝐼:dom 𝐼–1-1-onto→ran 𝐼 ↔ (𝐼 Fn dom 𝐼 ∧ ran 𝐼 = ran 𝐼 ∧ ∀𝑥 ∈ dom 𝐼∀𝑦 ∈ dom 𝐼((𝐼‘𝑥) = (𝐼‘𝑦) → 𝑥 = 𝑦)))
20	9, 10, 18, 19	syl3anbrc 1344	1 ⊢ ((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) → 𝐼:dom 𝐼–1-1-onto→ran 𝐼)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3044  {crab 3405   class class class wbr 5107  dom cdm 5638  ran crn 5639  Fun wfun 6505   Fn wfn 6506  –1-1-onto→wf1o 6510  ‘cfv 6511  Basecbs 17179  lecple 17227  HLchlt 39343  LHypclh 39978  DIsoAcdia 41022

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5234  ax-sep 5251  ax-nul 5261  ax-pow 5320  ax-pr 5387  ax-un 7711  ax-riotaBAD 38946

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-reu 3355  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-iin 4958  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-f1 6516  df-fo 6517  df-f1o 6518  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-1st 7968  df-2nd 7969  df-undef 8252  df-map 8801  df-proset 18255  df-poset 18274  df-plt 18289  df-lub 18305  df-glb 18306  df-join 18307  df-meet 18308  df-p0 18384  df-p1 18385  df-lat 18391  df-clat 18458  df-oposet 39169  df-ol 39171  df-oml 39172  df-covers 39259  df-ats 39260  df-atl 39291  df-cvlat 39315  df-hlat 39344  df-llines 39492  df-lplanes 39493  df-lvols 39494  df-lines 39495  df-psubsp 39497  df-pmap 39498  df-padd 39790  df-lhyp 39982  df-laut 39983  df-ldil 40098  df-ltrn 40099  df-trl 40153  df-disoa 41023

This theorem is referenced by:  diaclN  41044  diacnvclN  41045  dia1elN  41048  diainN  41051  diaintclN  41052  diasslssN  41053  docaclN  41118  diaocN  41119  doca3N  41121  diaf1oN  41124