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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 33885 | . 2 ⊢ Rel Bigcup | |
2 | mptrel 5680 | . 2 ⊢ Rel (𝑥 ∈ V ↦ ∪ 𝑥) | |
3 | eqcom 2743 | . . 3 ⊢ (∪ 𝑦 = 𝑧 ↔ 𝑧 = ∪ 𝑦) | |
4 | vex 3402 | . . . 4 ⊢ 𝑧 ∈ V | |
5 | 4 | brbigcup 33886 | . . 3 ⊢ (𝑦 Bigcup 𝑧 ↔ ∪ 𝑦 = 𝑧) |
6 | vex 3402 | . . . 4 ⊢ 𝑦 ∈ V | |
7 | eleq1w 2813 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ V ↔ 𝑦 ∈ V)) | |
8 | unieq 4816 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → ∪ 𝑥 = ∪ 𝑦) | |
9 | 8 | eqeq2d 2747 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑡 = ∪ 𝑥 ↔ 𝑡 = ∪ 𝑦)) |
10 | 7, 9 | anbi12d 634 | . . . . 5 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ V ∧ 𝑡 = ∪ 𝑥) ↔ (𝑦 ∈ V ∧ 𝑡 = ∪ 𝑦))) |
11 | 6 | biantrur 534 | . . . . 5 ⊢ (𝑡 = ∪ 𝑦 ↔ (𝑦 ∈ V ∧ 𝑡 = ∪ 𝑦)) |
12 | 10, 11 | bitr4di 292 | . . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ V ∧ 𝑡 = ∪ 𝑥) ↔ 𝑡 = ∪ 𝑦)) |
13 | eqeq1 2740 | . . . 4 ⊢ (𝑡 = 𝑧 → (𝑡 = ∪ 𝑦 ↔ 𝑧 = ∪ 𝑦)) | |
14 | df-mpt 5121 | . . . 4 ⊢ (𝑥 ∈ V ↦ ∪ 𝑥) = {〈𝑥, 𝑡〉 ∣ (𝑥 ∈ V ∧ 𝑡 = ∪ 𝑥)} | |
15 | 6, 4, 12, 13, 14 | brab 5409 | . . 3 ⊢ (𝑦(𝑥 ∈ V ↦ ∪ 𝑥)𝑧 ↔ 𝑧 = ∪ 𝑦) |
16 | 3, 5, 15 | 3bitr4i 306 | . 2 ⊢ (𝑦 Bigcup 𝑧 ↔ 𝑦(𝑥 ∈ V ↦ ∪ 𝑥)𝑧) |
17 | 1, 2, 16 | eqbrriv 5646 | 1 ⊢ Bigcup = (𝑥 ∈ V ↦ ∪ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 = wceq 1543 ∈ wcel 2112 Vcvv 3398 ∪ cuni 4805 class class class wbr 5039 ↦ cmpt 5120 Bigcup cbigcup 33822 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-sbc 3684 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-symdif 4143 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-mpt 5121 df-id 5440 df-eprel 5445 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-fo 6364 df-fv 6366 df-1st 7739 df-2nd 7740 df-txp 33842 df-bigcup 33846 |
This theorem is referenced by: fobigcup 33888 |
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