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Mirrors > Home > MPE Home > Th. List > Mathboxes > fobigcup | Structured version Visualization version GIF version |
Description: Bigcup maps the universe onto itself. (Contributed by Scott Fenton, 16-Apr-2012.) |
Ref | Expression |
---|---|
fobigcup | ⊢ Bigcup :V–onto→V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uniexg 7446 | . . . 4 ⊢ (𝑥 ∈ V → ∪ 𝑥 ∈ V) | |
2 | 1 | rgen 3116 | . . 3 ⊢ ∀𝑥 ∈ V ∪ 𝑥 ∈ V |
3 | dfbigcup2 33473 | . . . 4 ⊢ Bigcup = (𝑥 ∈ V ↦ ∪ 𝑥) | |
4 | 3 | mptfng 6459 | . . 3 ⊢ (∀𝑥 ∈ V ∪ 𝑥 ∈ V ↔ Bigcup Fn V) |
5 | 2, 4 | mpbi 233 | . 2 ⊢ Bigcup Fn V |
6 | 3 | rnmpt 5791 | . . 3 ⊢ ran Bigcup = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ 𝑥} |
7 | vex 3444 | . . . . 5 ⊢ 𝑦 ∈ V | |
8 | snex 5297 | . . . . . 6 ⊢ {𝑦} ∈ V | |
9 | 7 | unisn 4820 | . . . . . . 7 ⊢ ∪ {𝑦} = 𝑦 |
10 | 9 | eqcomi 2807 | . . . . . 6 ⊢ 𝑦 = ∪ {𝑦} |
11 | unieq 4811 | . . . . . . 7 ⊢ (𝑥 = {𝑦} → ∪ 𝑥 = ∪ {𝑦}) | |
12 | 11 | rspceeqv 3586 | . . . . . 6 ⊢ (({𝑦} ∈ V ∧ 𝑦 = ∪ {𝑦}) → ∃𝑥 ∈ V 𝑦 = ∪ 𝑥) |
13 | 8, 10, 12 | mp2an 691 | . . . . 5 ⊢ ∃𝑥 ∈ V 𝑦 = ∪ 𝑥 |
14 | 7, 13 | 2th 267 | . . . 4 ⊢ (𝑦 ∈ V ↔ ∃𝑥 ∈ V 𝑦 = ∪ 𝑥) |
15 | 14 | abbi2i 2929 | . . 3 ⊢ V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ 𝑥} |
16 | 6, 15 | eqtr4i 2824 | . 2 ⊢ ran Bigcup = V |
17 | df-fo 6330 | . 2 ⊢ ( Bigcup :V–onto→V ↔ ( Bigcup Fn V ∧ ran Bigcup = V)) | |
18 | 5, 16, 17 | mpbir2an 710 | 1 ⊢ Bigcup :V–onto→V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1538 ∈ wcel 2111 {cab 2776 ∀wral 3106 ∃wrex 3107 Vcvv 3441 {csn 4525 ∪ cuni 4800 ran crn 5520 Fn wfn 6319 –onto→wfo 6322 Bigcup cbigcup 33408 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-symdif 4169 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-eprel 5430 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-fo 6330 df-fv 6332 df-1st 7671 df-2nd 7672 df-txp 33428 df-bigcup 33432 |
This theorem is referenced by: fnbigcup 33475 |
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