Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > dfiunv2 | GIF version |
Description: Define double indexed union. (Contributed by FL, 6-Nov-2013.) |
Ref | Expression |
---|---|
dfiunv2 | ⊢ ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 𝐶 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 ∈ 𝐶} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-iun 3810 | . . . 4 ⊢ ∪ 𝑦 ∈ 𝐵 𝐶 = {𝑤 ∣ ∃𝑦 ∈ 𝐵 𝑤 ∈ 𝐶} | |
2 | 1 | a1i 9 | . . 3 ⊢ (𝑥 ∈ 𝐴 → ∪ 𝑦 ∈ 𝐵 𝐶 = {𝑤 ∣ ∃𝑦 ∈ 𝐵 𝑤 ∈ 𝐶}) |
3 | 2 | iuneq2i 3826 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 𝐶 = ∪ 𝑥 ∈ 𝐴 {𝑤 ∣ ∃𝑦 ∈ 𝐵 𝑤 ∈ 𝐶} |
4 | df-iun 3810 | . 2 ⊢ ∪ 𝑥 ∈ 𝐴 {𝑤 ∣ ∃𝑦 ∈ 𝐵 𝑤 ∈ 𝐶} = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ {𝑤 ∣ ∃𝑦 ∈ 𝐵 𝑤 ∈ 𝐶}} | |
5 | vex 2684 | . . . . 5 ⊢ 𝑧 ∈ V | |
6 | eleq1 2200 | . . . . . 6 ⊢ (𝑤 = 𝑧 → (𝑤 ∈ 𝐶 ↔ 𝑧 ∈ 𝐶)) | |
7 | 6 | rexbidv 2436 | . . . . 5 ⊢ (𝑤 = 𝑧 → (∃𝑦 ∈ 𝐵 𝑤 ∈ 𝐶 ↔ ∃𝑦 ∈ 𝐵 𝑧 ∈ 𝐶)) |
8 | 5, 7 | elab 2823 | . . . 4 ⊢ (𝑧 ∈ {𝑤 ∣ ∃𝑦 ∈ 𝐵 𝑤 ∈ 𝐶} ↔ ∃𝑦 ∈ 𝐵 𝑧 ∈ 𝐶) |
9 | 8 | rexbii 2440 | . . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ {𝑤 ∣ ∃𝑦 ∈ 𝐵 𝑤 ∈ 𝐶} ↔ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 ∈ 𝐶) |
10 | 9 | abbii 2253 | . 2 ⊢ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ {𝑤 ∣ ∃𝑦 ∈ 𝐵 𝑤 ∈ 𝐶}} = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 ∈ 𝐶} |
11 | 3, 4, 10 | 3eqtri 2162 | 1 ⊢ ∪ 𝑥 ∈ 𝐴 ∪ 𝑦 ∈ 𝐵 𝐶 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑧 ∈ 𝐶} |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ∈ wcel 1480 {cab 2123 ∃wrex 2415 ∪ ciun 3808 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-in 3072 df-ss 3079 df-iun 3810 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |