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| Mirrors > Home > MPE Home > Th. List > imaiun | Structured version Visualization version GIF version | ||
| Description: The image of an indexed union is the indexed union of the images. (Contributed by Mario Carneiro, 18-Jun-2014.) |
| Ref | Expression |
|---|---|
| imaiun | ⊢ (𝐴 “ ∪ 𝑥 ∈ 𝐵 𝐶) = ∪ 𝑥 ∈ 𝐵 (𝐴 “ 𝐶) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | rexcom4 3288 | . . . 4 ⊢ (∃𝑥 ∈ 𝐵 ∃𝑧(𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐴) ↔ ∃𝑧∃𝑥 ∈ 𝐵 (𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐴)) | |
| 2 | vex 3484 | . . . . . 6 ⊢ 𝑦 ∈ V | |
| 3 | 2 | elima3 6085 | . . . . 5 ⊢ (𝑦 ∈ (𝐴 “ 𝐶) ↔ ∃𝑧(𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐴)) |
| 4 | 3 | rexbii 3094 | . . . 4 ⊢ (∃𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 “ 𝐶) ↔ ∃𝑥 ∈ 𝐵 ∃𝑧(𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐴)) |
| 5 | eliun 4995 | . . . . . . 7 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝑧 ∈ 𝐶) | |
| 6 | 5 | anbi1i 624 | . . . . . 6 ⊢ ((𝑧 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐴) ↔ (∃𝑥 ∈ 𝐵 𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐴)) |
| 7 | r19.41v 3189 | . . . . . 6 ⊢ (∃𝑥 ∈ 𝐵 (𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐴) ↔ (∃𝑥 ∈ 𝐵 𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐴)) | |
| 8 | 6, 7 | bitr4i 278 | . . . . 5 ⊢ ((𝑧 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐴) ↔ ∃𝑥 ∈ 𝐵 (𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐴)) |
| 9 | 8 | exbii 1848 | . . . 4 ⊢ (∃𝑧(𝑧 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐴) ↔ ∃𝑧∃𝑥 ∈ 𝐵 (𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐴)) |
| 10 | 1, 4, 9 | 3bitr4ri 304 | . . 3 ⊢ (∃𝑧(𝑧 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐴) ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 “ 𝐶)) |
| 11 | 2 | elima3 6085 | . . 3 ⊢ (𝑦 ∈ (𝐴 “ ∪ 𝑥 ∈ 𝐵 𝐶) ↔ ∃𝑧(𝑧 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐴)) |
| 12 | eliun 4995 | . . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐵 (𝐴 “ 𝐶) ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 “ 𝐶)) | |
| 13 | 10, 11, 12 | 3bitr4i 303 | . 2 ⊢ (𝑦 ∈ (𝐴 “ ∪ 𝑥 ∈ 𝐵 𝐶) ↔ 𝑦 ∈ ∪ 𝑥 ∈ 𝐵 (𝐴 “ 𝐶)) |
| 14 | 13 | eqriv 2734 | 1 ⊢ (𝐴 “ ∪ 𝑥 ∈ 𝐵 𝐶) = ∪ 𝑥 ∈ 𝐵 (𝐴 “ 𝐶) |
| Colors of variables: wff setvar class |
| Syntax hints: ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ∃wrex 3070 〈cop 4632 ∪ ciun 4991 “ cima 5688 |
| 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-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 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-xp 5691 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 |
| This theorem is referenced by: imauni 7266 uniqs 8817 hsmexlem4 10469 hsmexlem5 10470 xkococnlem 23667 ismbf3d 25689 mbfimaopnlem 25690 i1fima 25713 i1fd 25716 itg1addlem5 25735 limciun 25929 sibfof 34342 eulerpartlemgh 34380 poimirlem30 37657 itg2addnclem2 37679 ftc1anclem6 37705 uniqsALTV 38330 smfresal 46803 |
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