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Mirrors > Home > MPE Home > Th. List > rniun | Structured version Visualization version 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 3256 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦〈𝑦, 𝑧〉 ∈ 𝐵 ↔ ∃𝑦∃𝑥 ∈ 𝐴 〈𝑦, 𝑧〉 ∈ 𝐵) | |
2 | vex 3234 | . . . . . 6 ⊢ 𝑧 ∈ V | |
3 | 2 | elrn2 5397 | . . . . 5 ⊢ (𝑧 ∈ ran 𝐵 ↔ ∃𝑦〈𝑦, 𝑧〉 ∈ 𝐵) |
4 | 3 | rexbii 3070 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ ran 𝐵 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦〈𝑦, 𝑧〉 ∈ 𝐵) |
5 | eliun 4556 | . . . . 5 ⊢ (〈𝑦, 𝑧〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 〈𝑦, 𝑧〉 ∈ 𝐵) | |
6 | 5 | exbii 1814 | . . . 4 ⊢ (∃𝑦〈𝑦, 𝑧〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑦∃𝑥 ∈ 𝐴 〈𝑦, 𝑧〉 ∈ 𝐵) |
7 | 1, 4, 6 | 3bitr4ri 293 | . . 3 ⊢ (∃𝑦〈𝑦, 𝑧〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ ran 𝐵) |
8 | 2 | elrn2 5397 | . . 3 ⊢ (𝑧 ∈ ran ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑦〈𝑦, 𝑧〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) |
9 | eliun 4556 | . . 3 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ran 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ ran 𝐵) | |
10 | 7, 8, 9 | 3bitr4i 292 | . 2 ⊢ (𝑧 ∈ ran ∪ 𝑥 ∈ 𝐴 𝐵 ↔ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ran 𝐵) |
11 | 10 | eqriv 2648 | 1 ⊢ ran ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 ran 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1523 ∃wex 1744 ∈ wcel 2030 ∃wrex 2942 〈cop 4216 ∪ ciun 4552 ran crn 5144 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-sep 4814 ax-nul 4822 ax-pr 4936 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ral 2946 df-rex 2947 df-rab 2950 df-v 3233 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-nul 3949 df-if 4120 df-sn 4211 df-pr 4213 df-op 4217 df-iun 4554 df-br 4686 df-opab 4746 df-cnv 5151 df-dm 5153 df-rn 5154 |
This theorem is referenced by: rnuni 5579 fun11iun 7168 cnextf 21917 iunrelexp0 38311 |
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