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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 31174 | . 2 ⊢ Rel Bigcup | |
2 | mptrel 5170 | . 2 ⊢ Rel (𝑥 ∈ V ↦ ∪ 𝑥) | |
3 | eqcom 2617 | . . 3 ⊢ (∪ 𝑦 = 𝑧 ↔ 𝑧 = ∪ 𝑦) | |
4 | vex 3176 | . . . 4 ⊢ 𝑧 ∈ V | |
5 | 4 | brbigcup 31175 | . . 3 ⊢ (𝑦 Bigcup 𝑧 ↔ ∪ 𝑦 = 𝑧) |
6 | vex 3176 | . . . 4 ⊢ 𝑦 ∈ V | |
7 | eleq1 2676 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ V ↔ 𝑦 ∈ V)) | |
8 | unieq 4380 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → ∪ 𝑥 = ∪ 𝑦) | |
9 | 8 | eqeq2d 2620 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑡 = ∪ 𝑥 ↔ 𝑡 = ∪ 𝑦)) |
10 | 7, 9 | anbi12d 743 | . . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ V ∧ 𝑡 = ∪ 𝑥) ↔ (𝑦 ∈ V ∧ 𝑡 = ∪ 𝑦))) |
11 | 6 | biantrur 526 | . . . . 5 ⊢ (𝑡 = ∪ 𝑦 ↔ (𝑦 ∈ V ∧ 𝑡 = ∪ 𝑦)) |
12 | 10, 11 | syl6bbr 277 | . . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ V ∧ 𝑡 = ∪ 𝑥) ↔ 𝑡 = ∪ 𝑦)) |
13 | eqeq1 2614 | . . . 4 ⊢ (𝑡 = 𝑧 → (𝑡 = ∪ 𝑦 ↔ 𝑧 = ∪ 𝑦)) | |
14 | df-mpt 4645 | . . . 4 ⊢ (𝑥 ∈ V ↦ ∪ 𝑥) = {〈𝑥, 𝑡〉 ∣ (𝑥 ∈ V ∧ 𝑡 = ∪ 𝑥)} | |
15 | 6, 4, 12, 13, 14 | brab 4923 | . . 3 ⊢ (𝑦(𝑥 ∈ V ↦ ∪ 𝑥)𝑧 ↔ 𝑧 = ∪ 𝑦) |
16 | 3, 5, 15 | 3bitr4i 291 | . 2 ⊢ (𝑦 Bigcup 𝑧 ↔ 𝑦(𝑥 ∈ V ↦ ∪ 𝑥)𝑧) |
17 | 1, 2, 16 | eqbrriv 5138 | 1 ⊢ Bigcup = (𝑥 ∈ V ↦ ∪ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 383 = wceq 1475 ∈ wcel 1977 Vcvv 3173 ∪ cuni 4372 class class class wbr 4583 ↦ cmpt 4643 Bigcup cbigcup 31110 |
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-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-symdif 3806 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-eprel 4949 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-fo 5810 df-fv 5812 df-1st 7059 df-2nd 7060 df-txp 31130 df-bigcup 31134 |
This theorem is referenced by: fobigcup 31177 |
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