Theorem pltfval 16782

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 3185	. . 3 ⊢ (𝐾 ∈ 𝐴 → 𝐾 ∈ V)
3		fveq2 6103	. . . . . 6 ⊢ (𝑝 = 𝐾 → (le‘𝑝) = (le‘𝐾))
4		pltval.l	. . . . . 6 ⊢ ≤ = (le‘𝐾)
5	3, 4	syl6eqr 2662	. . . . 5 ⊢ (𝑝 = 𝐾 → (le‘𝑝) = ≤ )
6	5	difeq1d 3689	. . . 4 ⊢ (𝑝 = 𝐾 → ((le‘𝑝) ∖ I ) = ( ≤ ∖ I ))
7		df-plt 16781	. . . 4 ⊢ lt = (𝑝 ∈ V ↦ ((le‘𝑝) ∖ I ))
8		fvex 6113	. . . . . 6 ⊢ (le‘𝐾) ∈ V
9	4, 8	eqeltri 2684	. . . . 5 ⊢ ≤ ∈ V
10		difexg 4735	. . . . 5 ⊢ ( ≤ ∈ V → ( ≤ ∖ I ) ∈ V)
11	9, 10	ax-mp 5	. . . 4 ⊢ ( ≤ ∖ I ) ∈ V
12	6, 7, 11	fvmpt 6191	. . 3 ⊢ (𝐾 ∈ V → (lt‘𝐾) = ( ≤ ∖ I ))
13	2, 12	syl 17	. 2 ⊢ (𝐾 ∈ 𝐴 → (lt‘𝐾) = ( ≤ ∖ I ))
14	1, 13	syl5eq 2656	1 ⊢ (𝐾 ∈ 𝐴 → < = ( ≤ ∖ I ))

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∖ cdif 3537   I cid 4948  ‘cfv 5804  lecple 15775  ltcplt 16764

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-iota 5768  df-fun 5806  df-fv 5812  df-plt 16781

This theorem is referenced by:  pltval  16783  opsrtoslem2  19306  relt  19780  oppglt  28985  xrslt  29007  submarchi  29071