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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfbigcup2 | Structured version Visualization version GIF version |
Description: Bigcup using maps-to notation. (Contributed by Scott Fenton, 16-Apr-2012.) |
Ref | Expression |
---|---|
dfbigcup2 | ⊢ Bigcup = (𝑥 ∈ V ↦ ∪ 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | relbigcup 35879 | . 2 ⊢ Rel Bigcup | |
2 | mptrel 5838 | . 2 ⊢ Rel (𝑥 ∈ V ↦ ∪ 𝑥) | |
3 | eqcom 2742 | . . 3 ⊢ (∪ 𝑦 = 𝑧 ↔ 𝑧 = ∪ 𝑦) | |
4 | vex 3482 | . . . 4 ⊢ 𝑧 ∈ V | |
5 | 4 | brbigcup 35880 | . . 3 ⊢ (𝑦 Bigcup 𝑧 ↔ ∪ 𝑦 = 𝑧) |
6 | vex 3482 | . . . 4 ⊢ 𝑦 ∈ V | |
7 | eleq1w 2822 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ V ↔ 𝑦 ∈ V)) | |
8 | unieq 4923 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → ∪ 𝑥 = ∪ 𝑦) | |
9 | 8 | eqeq2d 2746 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑡 = ∪ 𝑥 ↔ 𝑡 = ∪ 𝑦)) |
10 | 7, 9 | anbi12d 632 | . . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ V ∧ 𝑡 = ∪ 𝑥) ↔ (𝑦 ∈ V ∧ 𝑡 = ∪ 𝑦))) |
11 | 6 | biantrur 530 | . . . . 5 ⊢ (𝑡 = ∪ 𝑦 ↔ (𝑦 ∈ V ∧ 𝑡 = ∪ 𝑦)) |
12 | 10, 11 | bitr4di 289 | . . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ V ∧ 𝑡 = ∪ 𝑥) ↔ 𝑡 = ∪ 𝑦)) |
13 | eqeq1 2739 | . . . 4 ⊢ (𝑡 = 𝑧 → (𝑡 = ∪ 𝑦 ↔ 𝑧 = ∪ 𝑦)) | |
14 | df-mpt 5232 | . . . 4 ⊢ (𝑥 ∈ V ↦ ∪ 𝑥) = {〈𝑥, 𝑡〉 ∣ (𝑥 ∈ V ∧ 𝑡 = ∪ 𝑥)} | |
15 | 6, 4, 12, 13, 14 | brab 5553 | . . 3 ⊢ (𝑦(𝑥 ∈ V ↦ ∪ 𝑥)𝑧 ↔ 𝑧 = ∪ 𝑦) |
16 | 3, 5, 15 | 3bitr4i 303 | . 2 ⊢ (𝑦 Bigcup 𝑧 ↔ 𝑦(𝑥 ∈ V ↦ ∪ 𝑥)𝑧) |
17 | 1, 2, 16 | eqbrriv 5804 | 1 ⊢ Bigcup = (𝑥 ∈ V ↦ ∪ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 = wceq 1537 ∈ wcel 2106 Vcvv 3478 ∪ cuni 4912 class class class wbr 5148 ↦ cmpt 5231 Bigcup cbigcup 35816 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-symdif 4259 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-eprel 5589 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fo 6569 df-fv 6571 df-1st 8013 df-2nd 8014 df-txp 35836 df-bigcup 35840 |
This theorem is referenced by: fobigcup 35882 |
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