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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 3288 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦〈𝑦, 𝑧〉 ∈ 𝐵 ↔ ∃𝑦∃𝑥 ∈ 𝐴 〈𝑦, 𝑧〉 ∈ 𝐵) | |
| 2 | vex 3484 | . . . . . 6 ⊢ 𝑧 ∈ V | |
| 3 | 2 | elrn2 5903 | . . . . 5 ⊢ (𝑧 ∈ ran 𝐵 ↔ ∃𝑦〈𝑦, 𝑧〉 ∈ 𝐵) |
| 4 | 3 | rexbii 3094 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ ran 𝐵 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦〈𝑦, 𝑧〉 ∈ 𝐵) |
| 5 | eliun 4995 | . . . . 5 ⊢ (〈𝑦, 𝑧〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 〈𝑦, 𝑧〉 ∈ 𝐵) | |
| 6 | 5 | exbii 1848 | . . . 4 ⊢ (∃𝑦〈𝑦, 𝑧〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑦∃𝑥 ∈ 𝐴 〈𝑦, 𝑧〉 ∈ 𝐵) |
| 7 | 1, 4, 6 | 3bitr4ri 304 | . . 3 ⊢ (∃𝑦〈𝑦, 𝑧〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ ran 𝐵) |
| 8 | 2 | elrn2 5903 | . . 3 ⊢ (𝑧 ∈ ran ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑦〈𝑦, 𝑧〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) |
| 9 | eliun 4995 | . . 3 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ran 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ ran 𝐵) | |
| 10 | 7, 8, 9 | 3bitr4i 303 | . 2 ⊢ (𝑧 ∈ ran ∪ 𝑥 ∈ 𝐴 𝐵 ↔ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ran 𝐵) |
| 11 | 10 | eqriv 2734 | 1 ⊢ ran ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 ran 𝐵 |
| Colors of variables: wff setvar class |
| Syntax hints: = wceq 1540 ∃wex 1779 ∈ wcel 2108 ∃wrex 3070 〈cop 4632 ∪ ciun 4991 ran crn 5686 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-11 2157 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-iun 4993 df-br 5144 df-opab 5206 df-cnv 5693 df-dm 5695 df-rn 5696 |
| This theorem is referenced by: rnuni 6168 fiun 7967 f1iun 7968 cnextf 24074 iunrelexp0 43715 |
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