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Mirrors > Home > MPE Home > Th. List > dfiun2g | Structured version Visualization version GIF version |
Description: Alternate definition of indexed union when 𝐵 is a set. Definition 15(a) of [Suppes] p. 44. (Contributed by NM, 23-Mar-2006.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
Ref | Expression |
---|---|
dfiun2g | ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfra1 2925 | . . . . . 6 ⊢ Ⅎ𝑥∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 | |
2 | rsp 2913 | . . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → (𝑥 ∈ 𝐴 → 𝐵 ∈ 𝐶)) | |
3 | clel3g 3310 | . . . . . . . 8 ⊢ (𝐵 ∈ 𝐶 → (𝑧 ∈ 𝐵 ↔ ∃𝑦(𝑦 = 𝐵 ∧ 𝑧 ∈ 𝑦))) | |
4 | 2, 3 | syl6 34 | . . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → (𝑥 ∈ 𝐴 → (𝑧 ∈ 𝐵 ↔ ∃𝑦(𝑦 = 𝐵 ∧ 𝑧 ∈ 𝑦)))) |
5 | 4 | imp 444 | . . . . . 6 ⊢ ((∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ∧ 𝑥 ∈ 𝐴) → (𝑧 ∈ 𝐵 ↔ ∃𝑦(𝑦 = 𝐵 ∧ 𝑧 ∈ 𝑦))) |
6 | 1, 5 | rexbida 3029 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → (∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃𝑥 ∈ 𝐴 ∃𝑦(𝑦 = 𝐵 ∧ 𝑧 ∈ 𝑦))) |
7 | rexcom4 3198 | . . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑦(𝑦 = 𝐵 ∧ 𝑧 ∈ 𝑦) ↔ ∃𝑦∃𝑥 ∈ 𝐴 (𝑦 = 𝐵 ∧ 𝑧 ∈ 𝑦)) | |
8 | 6, 7 | syl6bb 275 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → (∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃𝑦∃𝑥 ∈ 𝐴 (𝑦 = 𝐵 ∧ 𝑧 ∈ 𝑦))) |
9 | r19.41v 3070 | . . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 (𝑦 = 𝐵 ∧ 𝑧 ∈ 𝑦) ↔ (∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ∧ 𝑧 ∈ 𝑦)) | |
10 | 9 | exbii 1764 | . . . . 5 ⊢ (∃𝑦∃𝑥 ∈ 𝐴 (𝑦 = 𝐵 ∧ 𝑧 ∈ 𝑦) ↔ ∃𝑦(∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ∧ 𝑧 ∈ 𝑦)) |
11 | exancom 1774 | . . . . 5 ⊢ (∃𝑦(∃𝑥 ∈ 𝐴 𝑦 = 𝐵 ∧ 𝑧 ∈ 𝑦) ↔ ∃𝑦(𝑧 ∈ 𝑦 ∧ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵)) | |
12 | 10, 11 | bitri 263 | . . . 4 ⊢ (∃𝑦∃𝑥 ∈ 𝐴 (𝑦 = 𝐵 ∧ 𝑧 ∈ 𝑦) ↔ ∃𝑦(𝑧 ∈ 𝑦 ∧ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵)) |
13 | 8, 12 | syl6bb 275 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → (∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃𝑦(𝑧 ∈ 𝑦 ∧ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵))) |
14 | eliun 4460 | . . 3 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵) | |
15 | eluniab 4383 | . . 3 ⊢ (𝑧 ∈ ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵} ↔ ∃𝑦(𝑧 ∈ 𝑦 ∧ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵)) | |
16 | 13, 14, 15 | 3bitr4g 302 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ 𝑧 ∈ ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵})) |
17 | 16 | eqrdv 2608 | 1 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 → ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 = 𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 {cab 2596 ∀wral 2896 ∃wrex 2897 ∪ cuni 4372 ∪ ciun 4455 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-v 3175 df-uni 4373 df-iun 4457 |
This theorem is referenced by: dfiun2 4490 dfiun3g 5299 iunexg 7035 uniqs 7694 ac6num 9184 iunopn 20528 pnrmopn 20957 cncmp 21005 ptcmplem3 21668 iunmbl 23128 voliun 23129 sigaclcuni 29508 sigaclcu2 29510 sigaclci 29522 measvunilem 29602 meascnbl 29609 carsgclctunlem3 29709 |
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