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Theorem pltfval 17561
Description: Value of the less-than relation. (Contributed by Mario Carneiro, 8-Feb-2015.)
Hypotheses
Ref Expression
pltval.l = (le‘𝐾)
pltval.s < = (lt‘𝐾)
Assertion
Ref Expression
pltfval (𝐾𝐴< = ( ∖ I ))

Proof of Theorem pltfval
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 pltval.s . 2 < = (lt‘𝐾)
2 elex 3517 . . 3 (𝐾𝐴𝐾 ∈ V)
3 fveq2 6666 . . . . . 6 (𝑝 = 𝐾 → (le‘𝑝) = (le‘𝐾))
4 pltval.l . . . . . 6 = (le‘𝐾)
53, 4syl6eqr 2878 . . . . 5 (𝑝 = 𝐾 → (le‘𝑝) = )
65difeq1d 4101 . . . 4 (𝑝 = 𝐾 → ((le‘𝑝) ∖ I ) = ( ∖ I ))
7 df-plt 17560 . . . 4 lt = (𝑝 ∈ V ↦ ((le‘𝑝) ∖ I ))
84fvexi 6680 . . . . 5 ∈ V
98difexi 5228 . . . 4 ( ∖ I ) ∈ V
106, 7, 9fvmpt 6764 . . 3 (𝐾 ∈ V → (lt‘𝐾) = ( ∖ I ))
112, 10syl 17 . 2 (𝐾𝐴 → (lt‘𝐾) = ( ∖ I ))
121, 11syl5eq 2872 1 (𝐾𝐴< = ( ∖ I ))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1530  wcel 2107  Vcvv 3499  cdif 3936   I cid 5457  cfv 6351  lecple 16564  ltcplt 17543
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-iota 6311  df-fun 6353  df-fv 6359  df-plt 17560
This theorem is referenced by:  pltval  17562  opsrtoslem2  20185  relt  20675  oppglt  30555  xrslt  30577  submarchi  30729
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