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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 3249 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦〈𝑦, 𝑧〉 ∈ 𝐵 ↔ ∃𝑦∃𝑥 ∈ 𝐴 〈𝑦, 𝑧〉 ∈ 𝐵) | |
2 | vex 3497 | . . . . . 6 ⊢ 𝑧 ∈ V | |
3 | 2 | elrn2 5820 | . . . . 5 ⊢ (𝑧 ∈ ran 𝐵 ↔ ∃𝑦〈𝑦, 𝑧〉 ∈ 𝐵) |
4 | 3 | rexbii 3247 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ ran 𝐵 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦〈𝑦, 𝑧〉 ∈ 𝐵) |
5 | eliun 4922 | . . . . 5 ⊢ (〈𝑦, 𝑧〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 〈𝑦, 𝑧〉 ∈ 𝐵) | |
6 | 5 | exbii 1844 | . . . 4 ⊢ (∃𝑦〈𝑦, 𝑧〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑦∃𝑥 ∈ 𝐴 〈𝑦, 𝑧〉 ∈ 𝐵) |
7 | 1, 4, 6 | 3bitr4ri 306 | . . 3 ⊢ (∃𝑦〈𝑦, 𝑧〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ ran 𝐵) |
8 | 2 | elrn2 5820 | . . 3 ⊢ (𝑧 ∈ ran ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑦〈𝑦, 𝑧〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) |
9 | eliun 4922 | . . 3 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ran 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ ran 𝐵) | |
10 | 7, 8, 9 | 3bitr4i 305 | . 2 ⊢ (𝑧 ∈ ran ∪ 𝑥 ∈ 𝐴 𝐵 ↔ 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ran 𝐵) |
11 | 10 | eqriv 2818 | 1 ⊢ ran ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 ran 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∃wex 1776 ∈ wcel 2110 ∃wrex 3139 〈cop 4572 ∪ ciun 4918 ran crn 5555 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-iun 4920 df-br 5066 df-opab 5128 df-cnv 5562 df-dm 5564 df-rn 5565 |
This theorem is referenced by: rnuni 6006 fiun 7643 f1iun 7644 cnextf 22673 iunrelexp0 40045 |
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