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Theorem dibdmN 37178
Description: Domain of the partial isomorphism A. (Contributed by NM, 8-Mar-2014.) (New usage is discouraged.)
Hypotheses
Ref Expression
dibfn.b 𝐵 = (Base‘𝐾)
dibfn.l = (le‘𝐾)
dibfn.h 𝐻 = (LHyp‘𝐾)
dibfn.i 𝐼 = ((DIsoB‘𝐾)‘𝑊)
Assertion
Ref Expression
dibdmN ((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 dibdmN
StepHypRef Expression
1 dibfn.b . . 3 𝐵 = (Base‘𝐾)
2 dibfn.l . . 3 = (le‘𝐾)
3 dibfn.h . . 3 𝐻 = (LHyp‘𝐾)
4 dibfn.i . . 3 𝐼 = ((DIsoB‘𝐾)‘𝑊)
51, 2, 3, 4dibfnN 37177 . 2 ((𝐾𝑉𝑊𝐻) → 𝐼 Fn {𝑥𝐵𝑥 𝑊})
6 fndm 6201 . 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 385   = wceq 1653  wcel 2157  {crab 3093   class class class wbr 4843  dom cdm 5312   Fn wfn 6096  cfv 6101  Basecbs 16184  lecple 16274  LHypclh 36005  DIsoBcdib 37159
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-rep 4964  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5097  ax-un 7183
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-ral 3094  df-rex 3095  df-reu 3096  df-rab 3098  df-v 3387  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-pw 4351  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-iun 4712  df-br 4844  df-opab 4906  df-mpt 4923  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-res 5324  df-ima 5325  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-fv 6109  df-disoa 37050  df-dib 37160
This theorem is referenced by:  dibglbN  37187  dibintclN  37188  dihglblem3N  37316
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