Theorem kur14lem1 35400

Description: Lemma for kur14 35410. (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 3959	. . 3 ⊢ (𝑁 = 𝐴 → (𝑁 ⊆ 𝑋 ↔ 𝐴 ⊆ 𝑋))
3	1, 2	mpbiri 258	. 2 ⊢ (𝑁 = 𝐴 → 𝑁 ⊆ 𝑋)
4		difeq2 4072	. . . 4 ⊢ (𝑁 = 𝐴 → (𝑋 ∖ 𝑁) = (𝑋 ∖ 𝐴))
5		fveq2 6834	. . . 4 ⊢ (𝑁 = 𝐴 → (𝐾‘𝑁) = (𝐾‘𝐴))
6	4, 5	preq12d 4698	. . 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 4777	. . . 4 ⊢ (((𝑋 ∖ 𝐴) ∈ 𝑇 ∧ (𝐾‘𝐴) ∈ 𝑇) → {(𝑋 ∖ 𝐴), (𝐾‘𝐴)} ⊆ 𝑇)
10	7, 8, 9	mp2an 692	. . 3 ⊢ {(𝑋 ∖ 𝐴), (𝐾‘𝐴)} ⊆ 𝑇
11	6, 10	eqsstrdi 3978	. 2 ⊢ (𝑁 = 𝐴 → {(𝑋 ∖ 𝑁), (𝐾‘𝑁)} ⊆ 𝑇)
12	3, 11	jca 511	1 ⊢ (𝑁 = 𝐴 → (𝑁 ⊆ 𝑋 ∧ {(𝑋 ∖ 𝑁), (𝐾‘𝑁)} ⊆ 𝑇))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113   ∖ cdif 3898   ⊆ wss 3901  {cpr 4582  ‘cfv 6492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-iota 6448  df-fv 6500

This theorem is referenced by:  kur14lem7  35406