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Mirrors > Home > MPE Home > Th. List > Mathboxes > heiborlem2 | Structured version Visualization version GIF version |
Description: Lemma for heibor 33925. 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 2819 | . . 3 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ (𝐹‘𝑛) ↔ 𝐴 ∈ (𝐹‘𝑛))) | |
4 | oveq1 6812 | . . . 4 ⊢ (𝑦 = 𝐴 → (𝑦𝐵𝑛) = (𝐴𝐵𝑛)) | |
5 | 4 | eleq1d 2816 | . . 3 ⊢ (𝑦 = 𝐴 → ((𝑦𝐵𝑛) ∈ 𝐾 ↔ (𝐴𝐵𝑛) ∈ 𝐾)) |
6 | 3, 5 | 3anbi23d 1543 | . 2 ⊢ (𝑦 = 𝐴 → ((𝑛 ∈ ℕ0 ∧ 𝑦 ∈ (𝐹‘𝑛) ∧ (𝑦𝐵𝑛) ∈ 𝐾) ↔ (𝑛 ∈ ℕ0 ∧ 𝐴 ∈ (𝐹‘𝑛) ∧ (𝐴𝐵𝑛) ∈ 𝐾))) |
7 | eleq1 2819 | . . 3 ⊢ (𝑛 = 𝐶 → (𝑛 ∈ ℕ0 ↔ 𝐶 ∈ ℕ0)) | |
8 | fveq2 6344 | . . . 4 ⊢ (𝑛 = 𝐶 → (𝐹‘𝑛) = (𝐹‘𝐶)) | |
9 | 8 | eleq2d 2817 | . . 3 ⊢ (𝑛 = 𝐶 → (𝐴 ∈ (𝐹‘𝑛) ↔ 𝐴 ∈ (𝐹‘𝐶))) |
10 | oveq2 6813 | . . . 4 ⊢ (𝑛 = 𝐶 → (𝐴𝐵𝑛) = (𝐴𝐵𝐶)) | |
11 | 10 | eleq1d 2816 | . . 3 ⊢ (𝑛 = 𝐶 → ((𝐴𝐵𝑛) ∈ 𝐾 ↔ (𝐴𝐵𝐶) ∈ 𝐾)) |
12 | 7, 9, 11 | 3anbi123d 1540 | . 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 5140 | 1 ⊢ (𝐴𝐺𝐶 ↔ (𝐶 ∈ ℕ0 ∧ 𝐴 ∈ (𝐹‘𝐶) ∧ (𝐴𝐵𝐶) ∈ 𝐾)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 196 ∧ w3a 1072 = wceq 1624 ∈ wcel 2131 {cab 2738 ∃wrex 3043 Vcvv 3332 ∩ cin 3706 ⊆ wss 3707 𝒫 cpw 4294 ∪ cuni 4580 class class class wbr 4796 {copab 4856 ‘cfv 6041 (class class class)co 6805 Fincfn 8113 ℕ0cn0 11476 MetOpencmopn 19930 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1863 ax-4 1878 ax-5 1980 ax-6 2046 ax-7 2082 ax-9 2140 ax-10 2160 ax-11 2175 ax-12 2188 ax-13 2383 ax-ext 2732 ax-sep 4925 ax-nul 4933 ax-pr 5047 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1627 df-ex 1846 df-nf 1851 df-sb 2039 df-eu 2603 df-mo 2604 df-clab 2739 df-cleq 2745 df-clel 2748 df-nfc 2883 df-rex 3048 df-rab 3051 df-v 3334 df-dif 3710 df-un 3712 df-in 3714 df-ss 3721 df-nul 4051 df-if 4223 df-sn 4314 df-pr 4316 df-op 4320 df-uni 4581 df-br 4797 df-opab 4857 df-iota 6004 df-fv 6049 df-ov 6808 |
This theorem is referenced by: heiborlem3 33917 heiborlem5 33919 heiborlem6 33920 heiborlem8 33922 heiborlem10 33924 |
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