Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > iinab | Structured version Visualization version GIF version |
Description: Indexed intersection of a class builder. (Contributed by NM, 6-Dec-2011.) |
Ref | Expression |
---|---|
iinab | ⊢ ∩ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝜑} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfcv 2974 | . . . 4 ⊢ Ⅎ𝑦𝐴 | |
2 | nfab1 2976 | . . . 4 ⊢ Ⅎ𝑦{𝑦 ∣ 𝜑} | |
3 | 1, 2 | nfiin 4941 | . . 3 ⊢ Ⅎ𝑦∩ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} |
4 | nfab1 2976 | . . 3 ⊢ Ⅎ𝑦{𝑦 ∣ ∀𝑥 ∈ 𝐴 𝜑} | |
5 | 3, 4 | cleqf 3007 | . 2 ⊢ (∩ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝜑} ↔ ∀𝑦(𝑦 ∈ ∩ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ↔ 𝑦 ∈ {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝜑})) |
6 | abid 2800 | . . . 4 ⊢ (𝑦 ∈ {𝑦 ∣ 𝜑} ↔ 𝜑) | |
7 | 6 | ralbii 3162 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝑦 ∈ {𝑦 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐴 𝜑) |
8 | eliin 4915 | . . . 4 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ {𝑦 ∣ 𝜑})) | |
9 | 8 | elv 3497 | . . 3 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ {𝑦 ∣ 𝜑}) |
10 | abid 2800 | . . 3 ⊢ (𝑦 ∈ {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝜑} ↔ ∀𝑥 ∈ 𝐴 𝜑) | |
11 | 7, 9, 10 | 3bitr4i 304 | . 2 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ↔ 𝑦 ∈ {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝜑}) |
12 | 5, 11 | mpgbir 1791 | 1 ⊢ ∩ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝜑} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 = wceq 1528 ∈ wcel 2105 {cab 2796 ∀wral 3135 Vcvv 3492 ∩ ciin 4911 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-v 3494 df-iin 4913 |
This theorem is referenced by: iinrab 4982 |
Copyright terms: Public domain | W3C validator |