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Theorem pltfval 18288
Description: Value of the less-than relation. (Contributed by Mario Carneiro, 8-Feb-2015.)
Hypotheses
Ref Expression
pltval.l ≀ = (leβ€˜πΎ)
pltval.s < = (ltβ€˜πΎ)
Assertion
Ref Expression
pltfval (𝐾 ∈ 𝐴 β†’ < = ( ≀ βˆ– I ))

Proof of Theorem pltfval
Dummy variable 𝑝 is distinct from all other variables.
StepHypRef Expression
1 pltval.s . 2 < = (ltβ€˜πΎ)
2 elex 3491 . . 3 (𝐾 ∈ 𝐴 β†’ 𝐾 ∈ V)
3 fveq2 6890 . . . . . 6 (𝑝 = 𝐾 β†’ (leβ€˜π‘) = (leβ€˜πΎ))
4 pltval.l . . . . . 6 ≀ = (leβ€˜πΎ)
53, 4eqtr4di 2788 . . . . 5 (𝑝 = 𝐾 β†’ (leβ€˜π‘) = ≀ )
65difeq1d 4120 . . . 4 (𝑝 = 𝐾 β†’ ((leβ€˜π‘) βˆ– I ) = ( ≀ βˆ– I ))
7 df-plt 18287 . . . 4 lt = (𝑝 ∈ V ↦ ((leβ€˜π‘) βˆ– I ))
84fvexi 6904 . . . . 5 ≀ ∈ V
98difexi 5327 . . . 4 ( ≀ βˆ– I ) ∈ V
106, 7, 9fvmpt 6997 . . 3 (𝐾 ∈ V β†’ (ltβ€˜πΎ) = ( ≀ βˆ– I ))
112, 10syl 17 . 2 (𝐾 ∈ 𝐴 β†’ (ltβ€˜πΎ) = ( ≀ βˆ– I ))
121, 11eqtrid 2782 1 (𝐾 ∈ 𝐴 β†’ < = ( ≀ βˆ– I ))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1539   ∈ wcel 2104  Vcvv 3472   βˆ– cdif 3944   I cid 5572  β€˜cfv 6542  lecple 17208  ltcplt 18265
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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-iota 6494  df-fun 6544  df-fv 6550  df-plt 18287
This theorem is referenced by:  pltval  18289  relt  21387  opsrtoslem2  21836  oppglt  32399  xrslt  32444  submarchi  32602
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