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| Description: Alternate definition of indexed union when 𝐵 is a set. (Contributed by Mario Carneiro, 31-Aug-2015.) | 
| Ref | Expression | 
|---|---|
| dfiun3g | ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | dfiun2g 5029 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}) | |
| 2 | eqid 2736 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
| 3 | 2 | rnmpt 5967 | . . 3 ⊢ ran (𝑥 ∈ 𝐴 ↦ 𝐵) = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} | 
| 4 | 3 | unieqi 4918 | . 2 ⊢ ∪ ran (𝑥 ∈ 𝐴 ↦ 𝐵) = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} | 
| 5 | 1, 4 | eqtr4di 2794 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ ran (𝑥 ∈ 𝐴 ↦ 𝐵)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 {cab 2713 ∀wral 3060 ∃wrex 3069 ∪ cuni 4906 ∪ ciun 4990 ↦ cmpt 5224 ran crn 5685 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-cnv 5692 df-dm 5694 df-rn 5695 | 
| This theorem is referenced by: dfiun3 5979 iunon 8380 onoviun 8384 gruiun 10840 tgiun 22987 acunirnmpt2f 32672 locfinreflem 33840 carsgclctunlem2 34322 pmeasadd 34328 saliunclf 46342 salexct3 46362 salgensscntex 46364 meadjiun 46486 omeiunle 46537 ovolval5lem2 46673 | 
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