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Mirrors > Home > MPE Home > Th. List > Mathboxes > kur14lem1 | Structured version Visualization version GIF version |
Description: Lemma for kur14 32458. (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 3992 | . . 3 ⊢ (𝑁 = 𝐴 → (𝑁 ⊆ 𝑋 ↔ 𝐴 ⊆ 𝑋)) | |
3 | 1, 2 | mpbiri 260 | . 2 ⊢ (𝑁 = 𝐴 → 𝑁 ⊆ 𝑋) |
4 | difeq2 4093 | . . . 4 ⊢ (𝑁 = 𝐴 → (𝑋 ∖ 𝑁) = (𝑋 ∖ 𝐴)) | |
5 | fveq2 6665 | . . . 4 ⊢ (𝑁 = 𝐴 → (𝐾‘𝑁) = (𝐾‘𝐴)) | |
6 | 4, 5 | preq12d 4671 | . . 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 4748 | . . . 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 4021 | . 2 ⊢ (𝑁 = 𝐴 → {(𝑋 ∖ 𝑁), (𝐾‘𝑁)} ⊆ 𝑇) |
12 | 3, 11 | jca 514 | 1 ⊢ (𝑁 = 𝐴 → (𝑁 ⊆ 𝑋 ∧ {(𝑋 ∖ 𝑁), (𝐾‘𝑁)} ⊆ 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ∖ cdif 3933 ⊆ wss 3936 {cpr 4563 ‘cfv 6350 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-rex 3144 df-rab 3147 df-v 3497 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4833 df-br 5060 df-iota 6309 df-fv 6358 |
This theorem is referenced by: kur14lem7 32454 |
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