Theorem pltfval 18496

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 3472	. . 3 ⊢ (𝐾 ∈ 𝐴 → 𝐾 ∈ V)
3		fveq2 6883	. . . . . 6 ⊢ (𝑝 = 𝐾 → (le‘𝑝) = (le‘𝐾))
4		pltval.l	. . . . . 6 ⊢ ≤ = (le‘𝐾)
5	3, 4	eqtr4di 2814	. . . . 5 ⊢ (𝑝 = 𝐾 → (le‘𝑝) = ≤ )
6	5	difeq1d 4073	. . . 4 ⊢ (𝑝 = 𝐾 → ((le‘𝑝) ∖ I ) = ( ≤ ∖ I ))
7		df-plt 18495	. . . 4 ⊢ lt = (𝑝 ∈ V ↦ ((le‘𝑝) ∖ I ))
8	4	fvexi 6897	. . . . 5 ⊢ ≤ ∈ V
9	8	difexi 5292	. . . 4 ⊢ ( ≤ ∖ I ) ∈ V
10	6, 7, 9	fvmpt 6991	. . 3 ⊢ (𝐾 ∈ V → (lt‘𝐾) = ( ≤ ∖ I ))
11	2, 10	syl 18	. 2 ⊢ (𝐾 ∈ 𝐴 → (lt‘𝐾) = ( ≤ ∖ I ))
12	1, 11	eqtrid 2808	1 ⊢ (𝐾 ∈ 𝐴 → < = ( ≤ ∖ I ))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  Vcvv 3451   ∖ cdif 3896   I cid 5545  ‘cfv 6537  lecple 17428  ltcplt 18475

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-iota 6493  df-fun 6539  df-fv 6545  df-plt 18495

This theorem is used by:  pltval  18497  oppglt  19575  relt  21914  opsrtoslem2  22358  xrslt  33561  submarchi  33740