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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 3225 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦〈𝑦, 𝑧〉 ∈ 𝐵 ↔ ∃𝑦∃𝑥 ∈ 𝐴 〈𝑦, 𝑧〉 ∈ 𝐵) | |
2 | vex 3203 | . . . . . 6 ⊢ 𝑧 ∈ V | |
3 | 2 | elrn2 5365 | . . . . 5 ⊢ (𝑧 ∈ ran 𝐵 ↔ ∃𝑦〈𝑦, 𝑧〉 ∈ 𝐵) |
4 | 3 | rexbii 3041 | . . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ ran 𝐵 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦〈𝑦, 𝑧〉 ∈ 𝐵) |
5 | eliun 4524 | . . . . 5 ⊢ (〈𝑦, 𝑧〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 〈𝑦, 𝑧〉 ∈ 𝐵) | |
6 | 5 | exbii 1774 | . . . 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 5365 | . . 3 ⊢ (𝑧 ∈ ran ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑦〈𝑦, 𝑧〉 ∈ ∪ 𝑥 ∈ 𝐴 𝐵) |
9 | eliun 4524 | . . 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 2619 | 1 ⊢ ran ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑥 ∈ 𝐴 ran 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ∃wex 1704 ∈ wcel 1990 ∃wrex 2913 〈cop 4183 ∪ ciun 4520 ran crn 5115 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-iun 4522 df-br 4654 df-opab 4713 df-cnv 5122 df-dm 5124 df-rn 5125 |
This theorem is referenced by: rnuni 5544 fun11iun 7126 cnextf 21870 iunrelexp0 37994 |
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