Theorem kur14lem1 34725

Description: Lemma for kur14 34735. (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 4002	. . 3 ⊢ (𝑁 = 𝐴 → (𝑁 ⊆ 𝑋 ↔ 𝐴 ⊆ 𝑋))
3	1, 2	mpbiri 258	. 2 ⊢ (𝑁 = 𝐴 → 𝑁 ⊆ 𝑋)
4		difeq2 4111	. . . 4 ⊢ (𝑁 = 𝐴 → (𝑋 ∖ 𝑁) = (𝑋 ∖ 𝐴))
5		fveq2 6885	. . . 4 ⊢ (𝑁 = 𝐴 → (𝐾‘𝑁) = (𝐾‘𝐴))
6	4, 5	preq12d 4740	. . 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 4819	. . . 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 4031	. 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 1533   ∈ wcel 2098   ∖ cdif 3940   ⊆ wss 3943  {cpr 4625  ‘cfv 6537

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 2697

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-iota 6489  df-fv 6545

This theorem is referenced by:  kur14lem7  34731