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| Mirrors > Home > MPE Home > Th. List > Mathboxes > heiborlem2 | Structured version Visualization version GIF version | ||
| Description: Lemma for heibor 37828. 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 2829 | . . 3 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ (𝐹‘𝑛) ↔ 𝐴 ∈ (𝐹‘𝑛))) | |
| 4 | oveq1 7438 | . . . 4 ⊢ (𝑦 = 𝐴 → (𝑦𝐵𝑛) = (𝐴𝐵𝑛)) | |
| 5 | 4 | eleq1d 2826 | . . 3 ⊢ (𝑦 = 𝐴 → ((𝑦𝐵𝑛) ∈ 𝐾 ↔ (𝐴𝐵𝑛) ∈ 𝐾)) |
| 6 | 3, 5 | 3anbi23d 1441 | . 2 ⊢ (𝑦 = 𝐴 → ((𝑛 ∈ ℕ0 ∧ 𝑦 ∈ (𝐹‘𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾) ↔ (𝑛 ∈ ℕ0 ∧ 𝐴 ∈ (𝐹‘𝑛) ∧ (𝐴𝐵𝑛) ∈ 𝐾))) |
| 7 | eleq1 2829 | . . 3 ⊢ (𝑛 = 𝐶 → (𝑛 ∈ ℕ0 ↔ 𝐶 ∈ ℕ0)) | |
| 8 | fveq2 6906 | . . . 4 ⊢ (𝑛 = 𝐶 → (𝐹‘𝑛) = (𝐹‘𝐶)) | |
| 9 | 8 | eleq2d 2827 | . . 3 ⊢ (𝑛 = 𝐶 → (𝐴 ∈ (𝐹‘𝑛) ↔ 𝐴 ∈ (𝐹‘𝐶))) |
| 10 | oveq2 7439 | . . . 4 ⊢ (𝑛 = 𝐶 → (𝐴𝐵𝑛) = (𝐴𝐵𝐶)) | |
| 11 | 10 | eleq1d 2826 | . . 3 ⊢ (𝑛 = 𝐶 → ((𝐴𝐵𝑛) ∈ 𝐾 ↔ (𝐴𝐵𝐶) ∈ 𝐾)) |
| 12 | 7, 9, 11 | 3anbi123d 1438 | . 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 5548 | 1 ⊢ (𝐴𝐺𝐶 ↔ (𝐶 ∈ ℕ0 ∧ 𝐴 ∈ (𝐹‘𝐶) ∧ (𝐴𝐵𝐶) ∈ 𝐾)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ↔ wb 206 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 {cab 2714 ∃wrex 3070 Vcvv 3480 ∩ cin 3950 ⊆ wss 3951 𝒫 cpw 4600 ∪ cuni 4907 class class class wbr 5143 {copab 5205 ‘cfv 6561 (class class class)co 7431 Fincfn 8985 ℕ0cn0 12526 MetOpencmopn 21354 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-iota 6514 df-fv 6569 df-ov 7434 |
| This theorem is referenced by: heiborlem3 37820 heiborlem5 37822 heiborlem6 37823 heiborlem8 37825 heiborlem10 37827 |
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