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 2751 | . . . 4 ⊢ Ⅎ𝑦𝐴 | |
2 | nfab1 2753 | . . . 4 ⊢ Ⅎ𝑦{𝑦 ∣ 𝜑} | |
3 | 1, 2 | nfiin 4485 | . . 3 ⊢ Ⅎ𝑦∩ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} |
4 | nfab1 2753 | . . 3 ⊢ Ⅎ𝑦{𝑦 ∣ ∀𝑥 ∈ 𝐴 𝜑} | |
5 | 3, 4 | cleqf 2776 | . 2 ⊢ (∩ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝜑} ↔ ∀𝑦(𝑦 ∈ ∩ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ↔ 𝑦 ∈ {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝜑})) |
6 | abid 2598 | . . . 4 ⊢ (𝑦 ∈ {𝑦 ∣ 𝜑} ↔ 𝜑) | |
7 | 6 | ralbii 2963 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝑦 ∈ {𝑦 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐴 𝜑) |
8 | vex 3176 | . . . 4 ⊢ 𝑦 ∈ V | |
9 | eliin 4461 | . . . 4 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ {𝑦 ∣ 𝜑})) | |
10 | 8, 9 | ax-mp 5 | . . 3 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ {𝑦 ∣ 𝜑}) |
11 | abid 2598 | . . 3 ⊢ (𝑦 ∈ {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝜑} ↔ ∀𝑥 ∈ 𝐴 𝜑) | |
12 | 7, 10, 11 | 3bitr4i 291 | . 2 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} ↔ 𝑦 ∈ {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝜑}) |
13 | 5, 12 | mpgbir 1717 | 1 ⊢ ∩ 𝑥 ∈ 𝐴 {𝑦 ∣ 𝜑} = {𝑦 ∣ ∀𝑥 ∈ 𝐴 𝜑} |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 = wceq 1475 ∈ wcel 1977 {cab 2596 ∀wral 2896 Vcvv 3173 ∩ ciin 4456 |
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-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-v 3175 df-iin 4458 |
This theorem is referenced by: iinrab 4518 |
Copyright terms: Public domain | W3C validator |