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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 3249 | . . . 4 ⊢ (∃𝑥 ∈ 𝐵 ∃𝑧(𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐴) ↔ ∃𝑧∃𝑥 ∈ 𝐵 (𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐴)) | |
2 | vex 3498 | . . . . . 6 ⊢ 𝑦 ∈ V | |
3 | 2 | elima3 5931 | . . . . 5 ⊢ (𝑦 ∈ (𝐴 “ 𝐶) ↔ ∃𝑧(𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐴)) |
4 | 3 | rexbii 3247 | . . . 4 ⊢ (∃𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 “ 𝐶) ↔ ∃𝑥 ∈ 𝐵 ∃𝑧(𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐴)) |
5 | eliun 4916 | . . . . . . 7 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝑧 ∈ 𝐶) | |
6 | 5 | anbi1i 625 | . . . . . 6 ⊢ ((𝑧 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐴) ↔ (∃𝑥 ∈ 𝐵 𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐴)) |
7 | r19.41v 3347 | . . . . . 6 ⊢ (∃𝑥 ∈ 𝐵 (𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐴) ↔ (∃𝑥 ∈ 𝐵 𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐴)) | |
8 | 6, 7 | bitr4i 280 | . . . . 5 ⊢ ((𝑧 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐴) ↔ ∃𝑥 ∈ 𝐵 (𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐴)) |
9 | 8 | exbii 1844 | . . . 4 ⊢ (∃𝑧(𝑧 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐴) ↔ ∃𝑧∃𝑥 ∈ 𝐵 (𝑧 ∈ 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐴)) |
10 | 1, 4, 9 | 3bitr4ri 306 | . . 3 ⊢ (∃𝑧(𝑧 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐴) ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 “ 𝐶)) |
11 | 2 | elima3 5931 | . . 3 ⊢ (𝑦 ∈ (𝐴 “ ∪ 𝑥 ∈ 𝐵 𝐶) ↔ ∃𝑧(𝑧 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 ∧ 〈𝑧, 𝑦〉 ∈ 𝐴)) |
12 | eliun 4916 | . . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐵 (𝐴 “ 𝐶) ↔ ∃𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 “ 𝐶)) | |
13 | 10, 11, 12 | 3bitr4i 305 | . 2 ⊢ (𝑦 ∈ (𝐴 “ ∪ 𝑥 ∈ 𝐵 𝐶) ↔ 𝑦 ∈ ∪ 𝑥 ∈ 𝐵 (𝐴 “ 𝐶)) |
14 | 13 | eqriv 2818 | 1 ⊢ (𝐴 “ ∪ 𝑥 ∈ 𝐵 𝐶) = ∪ 𝑥 ∈ 𝐵 (𝐴 “ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1533 ∃wex 1776 ∈ wcel 2110 ∃wrex 3139 〈cop 4567 ∪ ciun 4912 “ cima 5553 |
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 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322 |
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 3497 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-iun 4914 df-br 5060 df-opab 5122 df-xp 5556 df-cnv 5558 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 |
This theorem is referenced by: imauni 6999 uniqs 8351 hsmexlem4 9845 hsmexlem5 9846 xkococnlem 22261 ismbf3d 24249 mbfimaopnlem 24250 i1fima 24273 i1fd 24276 itg1addlem5 24295 limciun 24486 sibfof 31593 eulerpartlemgh 31631 poimirlem30 34916 itg2addnclem2 34938 ftc1anclem6 34966 uniqsALTV 35580 smfresal 43056 |
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