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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 33358 | . 2 ⊢ Rel Bigcup | |
2 | mptrel 5697 | . 2 ⊢ Rel (𝑥 ∈ V ↦ ∪ 𝑥) | |
3 | eqcom 2828 | . . 3 ⊢ (∪ 𝑦 = 𝑧 ↔ 𝑧 = ∪ 𝑦) | |
4 | vex 3497 | . . . 4 ⊢ 𝑧 ∈ V | |
5 | 4 | brbigcup 33359 | . . 3 ⊢ (𝑦 Bigcup 𝑧 ↔ ∪ 𝑦 = 𝑧) |
6 | vex 3497 | . . . 4 ⊢ 𝑦 ∈ V | |
7 | eleq1w 2895 | . . . . . 6 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ V ↔ 𝑦 ∈ V)) | |
8 | unieq 4849 | . . . . . . 7 ⊢ (𝑥 = 𝑦 → ∪ 𝑥 = ∪ 𝑦) | |
9 | 8 | eqeq2d 2832 | . . . . . 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 533 | . . . . 5 ⊢ (𝑡 = ∪ 𝑦 ↔ (𝑦 ∈ V ∧ 𝑡 = ∪ 𝑦)) |
12 | 10, 11 | syl6bbr 291 | . . . 4 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ V ∧ 𝑡 = ∪ 𝑥) ↔ 𝑡 = ∪ 𝑦)) |
13 | eqeq1 2825 | . . . 4 ⊢ (𝑡 = 𝑧 → (𝑡 = ∪ 𝑦 ↔ 𝑧 = ∪ 𝑦)) | |
14 | df-mpt 5147 | . . . 4 ⊢ (𝑥 ∈ V ↦ ∪ 𝑥) = {〈𝑥, 𝑡〉 ∣ (𝑥 ∈ V ∧ 𝑡 = ∪ 𝑥)} | |
15 | 6, 4, 12, 13, 14 | brab 5430 | . . 3 ⊢ (𝑦(𝑥 ∈ V ↦ ∪ 𝑥)𝑧 ↔ 𝑧 = ∪ 𝑦) |
16 | 3, 5, 15 | 3bitr4i 305 | . 2 ⊢ (𝑦 Bigcup 𝑧 ↔ 𝑦(𝑥 ∈ V ↦ ∪ 𝑥)𝑧) |
17 | 1, 2, 16 | eqbrriv 5664 | 1 ⊢ Bigcup = (𝑥 ∈ V ↦ ∪ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 = wceq 1537 ∈ wcel 2114 Vcvv 3494 ∪ cuni 4838 class class class wbr 5066 ↦ cmpt 5146 Bigcup cbigcup 33295 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-symdif 4219 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-eprel 5465 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-fo 6361 df-fv 6363 df-1st 7689 df-2nd 7690 df-txp 33315 df-bigcup 33319 |
This theorem is referenced by: fobigcup 33361 |
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