Theorem kur14lem1 33168

Description: Lemma for kur14 33178. (Contributed by Mario Carneiro, 17-Feb-2015.)

Hypotheses

Ref	Expression
kur14lem1.a	⊢ 𝐴 ⊆ 𝑋
kur14lem1.c	⊢ (𝑋 ∖ 𝐴) ∈ 𝑇
kur14lem1.k	⊢ (𝐾‘𝐴) ∈ 𝑇

Assertion

Ref	Expression
kur14lem1	⊢ (𝑁 = 𝐴 → (𝑁 ⊆ 𝑋 ∧ {(𝑋 ∖ 𝑁), (𝐾‘𝑁)} ⊆ 𝑇))

Proof of Theorem kur14lem1

Step	Hyp	Ref	Expression
1		kur14lem1.a	. . 3 ⊢ 𝐴 ⊆ 𝑋
2		sseq1 3946	. . 3 ⊢ (𝑁 = 𝐴 → (𝑁 ⊆ 𝑋 ↔ 𝐴 ⊆ 𝑋))
3	1, 2	mpbiri 257	. 2 ⊢ (𝑁 = 𝐴 → 𝑁 ⊆ 𝑋)
4		difeq2 4051	. . . 4 ⊢ (𝑁 = 𝐴 → (𝑋 ∖ 𝑁) = (𝑋 ∖ 𝐴))
5		fveq2 6774	. . . 4 ⊢ (𝑁 = 𝐴 → (𝐾‘𝑁) = (𝐾‘𝐴))
6	4, 5	preq12d 4677	. . 3 ⊢ (𝑁 = 𝐴 → {(𝑋 ∖ 𝑁), (𝐾‘𝑁)} = {(𝑋 ∖ 𝐴), (𝐾‘𝐴)})
7		kur14lem1.c	. . . 4 ⊢ (𝑋 ∖ 𝐴) ∈ 𝑇
8		kur14lem1.k	. . . 4 ⊢ (𝐾‘𝐴) ∈ 𝑇
9		prssi 4754	. . . 4 ⊢ (((𝑋 ∖ 𝐴) ∈ 𝑇 ∧ (𝐾‘𝐴) ∈ 𝑇) → {(𝑋 ∖ 𝐴), (𝐾‘𝐴)} ⊆ 𝑇)
10	7, 8, 9	mp2an 689	. . 3 ⊢ {(𝑋 ∖ 𝐴), (𝐾‘𝐴)} ⊆ 𝑇
11	6, 10	eqsstrdi 3975	. 2 ⊢ (𝑁 = 𝐴 → {(𝑋 ∖ 𝑁), (𝐾‘𝑁)} ⊆ 𝑇)
12	3, 11	jca 512	1 ⊢ (𝑁 = 𝐴 → (𝑁 ⊆ 𝑋 ∧ {(𝑋 ∖ 𝑁), (𝐾‘𝑁)} ⊆ 𝑇))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ∖ cdif 3884   ⊆ wss 3887  {cpr 4563  ‘cfv 6433

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-iota 6391  df-fv 6441

This theorem is referenced by:  kur14lem7  33174