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Mirrors > Home > MPE Home > Th. List > Mathboxes > kur14lem1 | Structured version Visualization version GIF version |
Description: Lemma for kur14 33175. (Contributed by Mario Carneiro, 17-Feb-2015.) |
Ref | Expression |
---|---|
kur14lem1.a | ⊢ 𝐴 ⊆ 𝑋 |
kur14lem1.c | ⊢ (𝑋 ∖ 𝐴) ∈ 𝑇 |
kur14lem1.k | ⊢ (𝐾‘𝐴) ∈ 𝑇 |
Ref | Expression |
---|---|
kur14lem1 | ⊢ (𝑁 = 𝐴 → (𝑁 ⊆ 𝑋 ∧ {(𝑋 ∖ 𝑁), (𝐾‘𝑁)} ⊆ 𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | kur14lem1.a | . . 3 ⊢ 𝐴 ⊆ 𝑋 | |
2 | sseq1 3947 | . . 3 ⊢ (𝑁 = 𝐴 → (𝑁 ⊆ 𝑋 ↔ 𝐴 ⊆ 𝑋)) | |
3 | 1, 2 | mpbiri 257 | . 2 ⊢ (𝑁 = 𝐴 → 𝑁 ⊆ 𝑋) |
4 | difeq2 4052 | . . . 4 ⊢ (𝑁 = 𝐴 → (𝑋 ∖ 𝑁) = (𝑋 ∖ 𝐴)) | |
5 | fveq2 6776 | . . . 4 ⊢ (𝑁 = 𝐴 → (𝐾‘𝑁) = (𝐾‘𝐴)) | |
6 | 4, 5 | preq12d 4679 | . . 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 4756 | . . . 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 3976 | . 2 ⊢ (𝑁 = 𝐴 → {(𝑋 ∖ 𝑁), (𝐾‘𝑁)} ⊆ 𝑇) |
12 | 3, 11 | jca 512 | 1 ⊢ (𝑁 = 𝐴 → (𝑁 ⊆ 𝑋 ∧ {(𝑋 ∖ 𝑁), (𝐾‘𝑁)} ⊆ 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ∖ cdif 3885 ⊆ wss 3888 {cpr 4565 ‘cfv 6435 |
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 3433 df-dif 3891 df-un 3893 df-in 3895 df-ss 3905 df-nul 4259 df-if 4462 df-sn 4564 df-pr 4566 df-op 4570 df-uni 4842 df-br 5077 df-iota 6393 df-fv 6443 |
This theorem is referenced by: kur14lem7 33171 |
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