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Mirrors > Home > ILE Home > Th. List > rniun | GIF version |
Description: The range of an indexed union. (Contributed by Mario Carneiro, 29-May-2015.) |
Ref | Expression |
---|---|
rniun | ⊢ ran ∪ x ∈ A B = ∪ x ∈ A ran B |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rexcom4 2571 | . . . 4 ⊢ (∃x ∈ A ∃y〈y, z〉 ∈ B ↔ ∃y∃x ∈ A 〈y, z〉 ∈ B) | |
2 | vex 2554 | . . . . . 6 ⊢ z ∈ V | |
3 | 2 | elrn2 4519 | . . . . 5 ⊢ (z ∈ ran B ↔ ∃y〈y, z〉 ∈ B) |
4 | 3 | rexbii 2325 | . . . 4 ⊢ (∃x ∈ A z ∈ ran B ↔ ∃x ∈ A ∃y〈y, z〉 ∈ B) |
5 | eliun 3652 | . . . . 5 ⊢ (〈y, z〉 ∈ ∪ x ∈ A B ↔ ∃x ∈ A 〈y, z〉 ∈ B) | |
6 | 5 | exbii 1493 | . . . 4 ⊢ (∃y〈y, z〉 ∈ ∪ x ∈ A B ↔ ∃y∃x ∈ A 〈y, z〉 ∈ B) |
7 | 1, 4, 6 | 3bitr4ri 202 | . . 3 ⊢ (∃y〈y, z〉 ∈ ∪ x ∈ A B ↔ ∃x ∈ A z ∈ ran B) |
8 | 2 | elrn2 4519 | . . 3 ⊢ (z ∈ ran ∪ x ∈ A B ↔ ∃y〈y, z〉 ∈ ∪ x ∈ A B) |
9 | eliun 3652 | . . 3 ⊢ (z ∈ ∪ x ∈ A ran B ↔ ∃x ∈ A z ∈ ran B) | |
10 | 7, 8, 9 | 3bitr4i 201 | . 2 ⊢ (z ∈ ran ∪ x ∈ A B ↔ z ∈ ∪ x ∈ A ran B) |
11 | 10 | eqriv 2034 | 1 ⊢ ran ∪ x ∈ A B = ∪ x ∈ A ran B |
Colors of variables: wff set class |
Syntax hints: = wceq 1242 ∃wex 1378 ∈ wcel 1390 ∃wrex 2301 〈cop 3370 ∪ ciun 3648 ran crn 4289 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-iun 3650 df-br 3756 df-opab 3810 df-cnv 4296 df-dm 4298 df-rn 4299 |
This theorem is referenced by: rnuni 4678 fun11iun 5090 |
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