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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 3270 | . . . 4 ⊢ (∃𝑥 ∈ 𝐵 ∃𝑧(𝑧 ∈ 𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴) ↔ ∃𝑧∃𝑥 ∈ 𝐵 (𝑧 ∈ 𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴)) | |
2 | vex 3448 | . . . . . 6 ⊢ 𝑦 ∈ V | |
3 | 2 | elima3 6021 | . . . . 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 4959 | . . . . . . 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 3182 | . . . . . 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 1851 | . . . 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 6021 | . . 3 ⊢ (𝑦 ∈ (𝐴 “ ∪ 𝑥 ∈ 𝐵 𝐶) ↔ ∃𝑧(𝑧 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 ∧ ⟨𝑧, 𝑦⟩ ∈ 𝐴)) |
12 | eliun 4959 | . . 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 2730 | 1 ⊢ (𝐴 “ ∪ 𝑥 ∈ 𝐵 𝐶) = ∪ 𝑥 ∈ 𝐵 (𝐴 “ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 397 = wceq 1542 ∃wex 1782 ∈ wcel 2107 ∃wrex 3070 ⟨cop 4593 ∪ ciun 4955 “ cima 5637 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-11 2155 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-iun 4957 df-br 5107 df-opab 5169 df-xp 5640 df-cnv 5642 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 |
This theorem is referenced by: imauni 7194 uniqs 8719 hsmexlem4 10370 hsmexlem5 10371 xkococnlem 23026 ismbf3d 25034 mbfimaopnlem 25035 i1fima 25058 i1fd 25061 itg1addlem5 25081 limciun 25274 sibfof 32997 eulerpartlemgh 33035 poimirlem30 36154 itg2addnclem2 36176 ftc1anclem6 36202 uniqsALTV 36836 smfresal 45115 |
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