Theorem pltfval 18375

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 3478	. . 3 ⊢ (𝐾 ∈ 𝐴 → 𝐾 ∈ V)
3		fveq2 6871	. . . . . 6 ⊢ (𝑝 = 𝐾 → (le‘𝑝) = (le‘𝐾))
4		pltval.l	. . . . . 6 ⊢ ≤ = (le‘𝐾)
5	3, 4	eqtr4di 2818	. . . . 5 ⊢ (𝑝 = 𝐾 → (le‘𝑝) = ≤ )
6	5	difeq1d 4082	. . . 4 ⊢ (𝑝 = 𝐾 → ((le‘𝑝) ∖ I ) = ( ≤ ∖ I ))
7		df-plt 18374	. . . 4 ⊢ lt = (𝑝 ∈ V ↦ ((le‘𝑝) ∖ I ))
8	4	fvexi 6885	. . . . 5 ⊢ ≤ ∈ V
9	8	difexi 5291	. . . 4 ⊢ ( ≤ ∖ I ) ∈ V
10	6, 7, 9	fvmpt 6979	. . 3 ⊢ (𝐾 ∈ V → (lt‘𝐾) = ( ≤ ∖ I ))
11	2, 10	syl 18	. 2 ⊢ (𝐾 ∈ 𝐴 → (lt‘𝐾) = ( ≤ ∖ I ))
12	1, 11	eqtrid 2812	1 ⊢ (𝐾 ∈ 𝐴 → < = ( ≤ ∖ I ))

Syntax hints:   → wi 4   = wceq 1563   ∈ wcel 2145  Vcvv 3457   ∖ cdif 3904   I cid 5546  ‘cfv 6525  lecple 17307  ltcplt 18354

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-iota 6481  df-fun 6527  df-fv 6533  df-plt 18374

This theorem is referenced by:  pltval  18376  oppglt  19429  relt  21725  opsrtoslem2  22167  xrslt  33240  submarchi  33419