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Theorem diadm 36838
Description: Domain of the partial isomorphism A. (Contributed by NM, 3-Dec-2013.)
Hypotheses
Ref Expression
diafn.b 𝐵 = (Base‘𝐾)
diafn.l = (le‘𝐾)
diafn.h 𝐻 = (LHyp‘𝐾)
diafn.i 𝐼 = ((DIsoA‘𝐾)‘𝑊)
Assertion
Ref Expression
diadm ((𝐾𝑉𝑊𝐻) → dom 𝐼 = {𝑥𝐵𝑥 𝑊})
Distinct variable groups:   𝑥,   𝑥,𝐵   𝑥,𝐾   𝑥,𝑊
Allowed substitution hints:   𝐻(𝑥)   𝐼(𝑥)   𝑉(𝑥)

Proof of Theorem diadm
StepHypRef Expression
1 diafn.b . . 3 𝐵 = (Base‘𝐾)
2 diafn.l . . 3 = (le‘𝐾)
3 diafn.h . . 3 𝐻 = (LHyp‘𝐾)
4 diafn.i . . 3 𝐼 = ((DIsoA‘𝐾)‘𝑊)
51, 2, 3, 4diafn 36837 . 2 ((𝐾𝑉𝑊𝐻) → 𝐼 Fn {𝑥𝐵𝑥 𝑊})
6 fndm 6130 . 2 (𝐼 Fn {𝑥𝐵𝑥 𝑊} → dom 𝐼 = {𝑥𝐵𝑥 𝑊})
75, 6syl 17 1 ((𝐾𝑉𝑊𝐻) → dom 𝐼 = {𝑥𝐵𝑥 𝑊})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1630  wcel 2144  {crab 3064   class class class wbr 4784  dom cdm 5249   Fn wfn 6026  cfv 6031  Basecbs 16063  lecple 16155  LHypclh 35785  DIsoAcdia 36831
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-disoa 36832
This theorem is referenced by:  diaeldm  36839  diaglbN  36858  diaintclN  36861  dibfnN  36959  dibglbN  36969
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