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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 34449 | . 2 ⊢ Rel Bigcup | |
2 | mptrel 5779 | . 2 ⊢ Rel (𝑥 ∈ V ↦ ∪ 𝑥) | |
3 | eqcom 2743 | . . 3 ⊢ (∪ 𝑦 = 𝑧 ↔ 𝑧 = ∪ 𝑦) | |
4 | vex 3447 | . . . 4 ⊢ 𝑧 ∈ V | |
5 | 4 | brbigcup 34450 | . . 3 ⊢ (𝑦 Bigcup 𝑧 ↔ ∪ 𝑦 = 𝑧) |
6 | vex 3447 | . . . 4 ⊢ 𝑦 ∈ V | |
7 | eleq1w 2820 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ V ↔ 𝑦 ∈ V)) | |
8 | unieq 4874 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → ∪ 𝑥 = ∪ 𝑦) | |
9 | 8 | eqeq2d 2747 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑡 = ∪ 𝑥 ↔ 𝑡 = ∪ 𝑦)) |
10 | 7, 9 | anbi12d 631 | . . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ V ∧ 𝑡 = ∪ 𝑥) ↔ (𝑦 ∈ V ∧ 𝑡 = ∪ 𝑦))) |
11 | 6 | biantrur 531 | . . . . 5 ⊢ (𝑡 = ∪ 𝑦 ↔ (𝑦 ∈ V ∧ 𝑡 = ∪ 𝑦)) |
12 | 10, 11 | bitr4di 288 | . . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ V ∧ 𝑡 = ∪ 𝑥) ↔ 𝑡 = ∪ 𝑦)) |
13 | eqeq1 2740 | . . . 4 ⊢ (𝑡 = 𝑧 → (𝑡 = ∪ 𝑦 ↔ 𝑧 = ∪ 𝑦)) | |
14 | df-mpt 5187 | . . . 4 ⊢ (𝑥 ∈ V ↦ ∪ 𝑥) = {〈𝑥, 𝑡〉 ∣ (𝑥 ∈ V ∧ 𝑡 = ∪ 𝑥)} | |
15 | 6, 4, 12, 13, 14 | brab 5498 | . . 3 ⊢ (𝑦(𝑥 ∈ V ↦ ∪ 𝑥)𝑧 ↔ 𝑧 = ∪ 𝑦) |
16 | 3, 5, 15 | 3bitr4i 302 | . 2 ⊢ (𝑦 Bigcup 𝑧 ↔ 𝑦(𝑥 ∈ V ↦ ∪ 𝑥)𝑧) |
17 | 1, 2, 16 | eqbrriv 5745 | 1 ⊢ Bigcup = (𝑥 ∈ V ↦ ∪ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 = wceq 1541 ∈ wcel 2106 Vcvv 3443 ∪ cuni 4863 class class class wbr 5103 ↦ cmpt 5186 Bigcup cbigcup 34386 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5254 ax-nul 5261 ax-pow 5318 ax-pr 5382 ax-un 7668 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3406 df-v 3445 df-dif 3911 df-un 3913 df-in 3915 df-ss 3925 df-symdif 4200 df-nul 4281 df-if 4485 df-pw 4560 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4864 df-br 5104 df-opab 5166 df-mpt 5187 df-id 5529 df-eprel 5535 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-iota 6445 df-fun 6495 df-fn 6496 df-f 6497 df-fo 6499 df-fv 6501 df-1st 7917 df-2nd 7918 df-txp 34406 df-bigcup 34410 |
This theorem is referenced by: fobigcup 34452 |
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