Theorem pltfval 17964

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	⊢ (𝐾 ∈ 𝐴 → < = ( ≤ ∖ I ))

Proof of Theorem pltfval

Dummy variable 𝑝 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pltval.s	. 2 ⊢ < = (lt‘𝐾)
2		elex 3440	. . 3 ⊢ (𝐾 ∈ 𝐴 → 𝐾 ∈ V)
3		fveq2 6756	. . . . . 6 ⊢ (𝑝 = 𝐾 → (le‘𝑝) = (le‘𝐾))
4		pltval.l	. . . . . 6 ⊢ ≤ = (le‘𝐾)
5	3, 4	eqtr4di 2797	. . . . 5 ⊢ (𝑝 = 𝐾 → (le‘𝑝) = ≤ )
6	5	difeq1d 4052	. . . 4 ⊢ (𝑝 = 𝐾 → ((le‘𝑝) ∖ I ) = ( ≤ ∖ I ))
7		df-plt 17963	. . . 4 ⊢ lt = (𝑝 ∈ V ↦ ((le‘𝑝) ∖ I ))
8	4	fvexi 6770	. . . . 5 ⊢ ≤ ∈ V
9	8	difexi 5247	. . . 4 ⊢ ( ≤ ∖ I ) ∈ V
10	6, 7, 9	fvmpt 6857	. . 3 ⊢ (𝐾 ∈ V → (lt‘𝐾) = ( ≤ ∖ I ))
11	2, 10	syl 17	. 2 ⊢ (𝐾 ∈ 𝐴 → (lt‘𝐾) = ( ≤ ∖ I ))
12	1, 11	eqtrid 2790	1 ⊢ (𝐾 ∈ 𝐴 → < = ( ≤ ∖ I ))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  Vcvv 3422   ∖ cdif 3880   I cid 5479  ‘cfv 6418  lecple 16895  ltcplt 17941

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-iota 6376  df-fun 6420  df-fv 6426  df-plt 17963

This theorem is referenced by:  pltval  17965  relt  20732  opsrtoslem2  21173  oppglt  31142  xrslt  31187  submarchi  31342