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	⊢ (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 3459	. . 3 ⊢ (𝐾 ∈ 𝐴 → 𝐾 ∈ V)
3		fveq2 6645	. . . . . 6 ⊢ (𝑝 = 𝐾 → (le‘𝑝) = (le‘𝐾))
4		pltval.l	. . . . . 6 ⊢ ≤ = (le‘𝐾)
5	3, 4	eqtr4di 2851	. . . . 5 ⊢ (𝑝 = 𝐾 → (le‘𝑝) = ≤ )
6	5	difeq1d 4049	. . . 4 ⊢ (𝑝 = 𝐾 → ((le‘𝑝) ∖ I ) = ( ≤ ∖ I ))
7		df-plt 17560	. . . 4 ⊢ lt = (𝑝 ∈ V ↦ ((le‘𝑝) ∖ I ))
8	4	fvexi 6659	. . . . 5 ⊢ ≤ ∈ V
9	8	difexi 5196	. . . 4 ⊢ ( ≤ ∖ I ) ∈ V
10	6, 7, 9	fvmpt 6745	. . 3 ⊢ (𝐾 ∈ V → (lt‘𝐾) = ( ≤ ∖ I ))
11	2, 10	syl 17	. 2 ⊢ (𝐾 ∈ 𝐴 → (lt‘𝐾) = ( ≤ ∖ I ))
12	1, 11	syl5eq 2845	1 ⊢ (𝐾 ∈ 𝐴 → < = ( ≤ ∖ I ))

Syntax hints:   → wi 4   = wceq 1538   ∈ wcel 2111  Vcvv 3441   ∖ cdif 3878   I cid 5424  ‘cfv 6324  lecple 16564  ltcplt 17543

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332  df-plt 17560

This theorem is referenced by:  pltval  17562  relt  20304  opsrtoslem2  20724  oppglt  30667  xrslt  30710  submarchi  30865