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Mirrors > Home > MPE Home > Th. List > Mathboxes > heiborlem2 | Structured version Visualization version GIF version |
Description: Lemma for heibor 32790. Substitutions for the set 𝐺. (Contributed by Jeff Madsen, 23-Jan-2014.) |
Ref | Expression |
---|---|
heibor.1 | ⊢ 𝐽 = (MetOpen‘𝐷) |
heibor.3 | ⊢ 𝐾 = {𝑢 ∣ ¬ ∃𝑣 ∈ (𝒫 𝑈 ∩ Fin)𝑢 ⊆ ∪ 𝑣} |
heibor.4 | ⊢ 𝐺 = {〈𝑦, 𝑛〉 ∣ (𝑛 ∈ ℕ0 ∧ 𝑦 ∈ (𝐹‘𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)} |
heiborlem2.5 | ⊢ 𝐴 ∈ V |
heiborlem2.6 | ⊢ 𝐶 ∈ V |
Ref | Expression |
---|---|
heiborlem2 | ⊢ (𝐴𝐺𝐶 ↔ (𝐶 ∈ ℕ0 ∧ 𝐴 ∈ (𝐹‘𝐶) ∧ (𝐴𝐵𝐶) ∈ 𝐾)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | heiborlem2.5 | . 2 ⊢ 𝐴 ∈ V | |
2 | heiborlem2.6 | . 2 ⊢ 𝐶 ∈ V | |
3 | eleq1 2676 | . . 3 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ (𝐹‘𝑛) ↔ 𝐴 ∈ (𝐹‘𝑛))) | |
4 | oveq1 6556 | . . . 4 ⊢ (𝑦 = 𝐴 → (𝑦𝐵𝑛) = (𝐴𝐵𝑛)) | |
5 | 4 | eleq1d 2672 | . . 3 ⊢ (𝑦 = 𝐴 → ((𝑦𝐵𝑛) ∈ 𝐾 ↔ (𝐴𝐵𝑛) ∈ 𝐾)) |
6 | 3, 5 | 3anbi23d 1394 | . 2 ⊢ (𝑦 = 𝐴 → ((𝑛 ∈ ℕ0 ∧ 𝑦 ∈ (𝐹‘𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾) ↔ (𝑛 ∈ ℕ0 ∧ 𝐴 ∈ (𝐹‘𝑛) ∧ (𝐴𝐵𝑛) ∈ 𝐾))) |
7 | eleq1 2676 | . . 3 ⊢ (𝑛 = 𝐶 → (𝑛 ∈ ℕ0 ↔ 𝐶 ∈ ℕ0)) | |
8 | fveq2 6103 | . . . 4 ⊢ (𝑛 = 𝐶 → (𝐹‘𝑛) = (𝐹‘𝐶)) | |
9 | 8 | eleq2d 2673 | . . 3 ⊢ (𝑛 = 𝐶 → (𝐴 ∈ (𝐹‘𝑛) ↔ 𝐴 ∈ (𝐹‘𝐶))) |
10 | oveq2 6557 | . . . 4 ⊢ (𝑛 = 𝐶 → (𝐴𝐵𝑛) = (𝐴𝐵𝐶)) | |
11 | 10 | eleq1d 2672 | . . 3 ⊢ (𝑛 = 𝐶 → ((𝐴𝐵𝑛) ∈ 𝐾 ↔ (𝐴𝐵𝐶) ∈ 𝐾)) |
12 | 7, 9, 11 | 3anbi123d 1391 | . 2 ⊢ (𝑛 = 𝐶 → ((𝑛 ∈ ℕ0 ∧ 𝐴 ∈ (𝐹‘𝑛) ∧ (𝐴𝐵𝑛) ∈ 𝐾) ↔ (𝐶 ∈ ℕ0 ∧ 𝐴 ∈ (𝐹‘𝐶) ∧ (𝐴𝐵𝐶) ∈ 𝐾))) |
13 | heibor.4 | . 2 ⊢ 𝐺 = {〈𝑦, 𝑛〉 ∣ (𝑛 ∈ ℕ0 ∧ 𝑦 ∈ (𝐹‘𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾)} | |
14 | 1, 2, 6, 12, 13 | brab 4923 | 1 ⊢ (𝐴𝐺𝐶 ↔ (𝐶 ∈ ℕ0 ∧ 𝐴 ∈ (𝐹‘𝐶) ∧ (𝐴𝐵𝐶) ∈ 𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 195 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 {cab 2596 ∃wrex 2897 Vcvv 3173 ∩ cin 3539 ⊆ wss 3540 𝒫 cpw 4108 ∪ cuni 4372 class class class wbr 4583 {copab 4642 ‘cfv 5804 (class class class)co 6549 Fincfn 7841 ℕ0cn0 11169 MetOpencmopn 19557 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-iota 5768 df-fv 5812 df-ov 6552 |
This theorem is referenced by: heiborlem3 32782 heiborlem5 32784 heiborlem6 32785 heiborlem8 32787 heiborlem10 32789 |
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