Theorem kur14lem1 34845

Description: Lemma for kur14 34855. (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 3997	. . 3 ⊢ (𝑁 = 𝐴 → (𝑁 ⊆ 𝑋 ↔ 𝐴 ⊆ 𝑋))
3	1, 2	mpbiri 257	. 2 ⊢ (𝑁 = 𝐴 → 𝑁 ⊆ 𝑋)
4		difeq2 4106	. . . 4 ⊢ (𝑁 = 𝐴 → (𝑋 ∖ 𝑁) = (𝑋 ∖ 𝐴))
5		fveq2 6890	. . . 4 ⊢ (𝑁 = 𝐴 → (𝐾‘𝑁) = (𝐾‘𝐴))
6	4, 5	preq12d 4739	. . 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 4818	. . . 4 ⊢ (((𝑋 ∖ 𝐴) ∈ 𝑇 ∧ (𝐾‘𝐴) ∈ 𝑇) → {(𝑋 ∖ 𝐴), (𝐾‘𝐴)} ⊆ 𝑇)
10	7, 8, 9	mp2an 690	. . 3 ⊢ {(𝑋 ∖ 𝐴), (𝐾‘𝐴)} ⊆ 𝑇
11	6, 10	eqsstrdi 4026	. 2 ⊢ (𝑁 = 𝐴 → {(𝑋 ∖ 𝑁), (𝐾‘𝑁)} ⊆ 𝑇)
12	3, 11	jca 510	1 ⊢ (𝑁 = 𝐴 → (𝑁 ⊆ 𝑋 ∧ {(𝑋 ∖ 𝑁), (𝐾‘𝑁)} ⊆ 𝑇))

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1533   ∈ wcel 2098   ∖ cdif 3936   ⊆ wss 3939  {cpr 4624  ‘cfv 6541

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-nul 4317  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-uni 4902  df-br 5142  df-iota 6493  df-fv 6549

This theorem is referenced by:  kur14lem7  34851