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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 7478 | . . . 4 ⊢ (𝑥 ∈ V → ∪ 𝑥 ∈ V) | |
2 | 1 | rgen 3063 | . . 3 ⊢ ∀𝑥 ∈ V ∪ 𝑥 ∈ V |
3 | dfbigcup2 33831 | . . . 4 ⊢ Bigcup = (𝑥 ∈ V ↦ ∪ 𝑥) | |
4 | 3 | mptfng 6470 | . . 3 ⊢ (∀𝑥 ∈ V ∪ 𝑥 ∈ V ↔ Bigcup Fn V) |
5 | 2, 4 | mpbi 233 | . 2 ⊢ Bigcup Fn V |
6 | 3 | rnmpt 5792 | . . 3 ⊢ ran Bigcup = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ 𝑥} |
7 | vex 3401 | . . . . 5 ⊢ 𝑦 ∈ V | |
8 | snex 5295 | . . . . . 6 ⊢ {𝑦} ∈ V | |
9 | 7 | unisn 4815 | . . . . . . 7 ⊢ ∪ {𝑦} = 𝑦 |
10 | 9 | eqcomi 2747 | . . . . . 6 ⊢ 𝑦 = ∪ {𝑦} |
11 | unieq 4804 | . . . . . . 7 ⊢ (𝑥 = {𝑦} → ∪ 𝑥 = ∪ {𝑦}) | |
12 | 11 | rspceeqv 3539 | . . . . . 6 ⊢ (({𝑦} ∈ V ∧ 𝑦 = ∪ {𝑦}) → ∃𝑥 ∈ V 𝑦 = ∪ 𝑥) |
13 | 8, 10, 12 | mp2an 692 | . . . . 5 ⊢ ∃𝑥 ∈ V 𝑦 = ∪ 𝑥 |
14 | 7, 13 | 2th 267 | . . . 4 ⊢ (𝑦 ∈ V ↔ ∃𝑥 ∈ V 𝑦 = ∪ 𝑥) |
15 | 14 | abbi2i 2871 | . . 3 ⊢ V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ 𝑥} |
16 | 6, 15 | eqtr4i 2764 | . 2 ⊢ ran Bigcup = V |
17 | df-fo 6339 | . 2 ⊢ ( Bigcup :V–onto→V ↔ ( Bigcup Fn V ∧ ran Bigcup = V)) | |
18 | 5, 16, 17 | mpbir2an 711 | 1 ⊢ Bigcup :V–onto→V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∈ wcel 2113 {cab 2716 ∀wral 3053 ∃wrex 3054 Vcvv 3397 {csn 4513 ∪ cuni 4793 ran crn 5520 Fn wfn 6328 –onto→wfo 6331 Bigcup cbigcup 33766 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 ax-sep 5164 ax-nul 5171 ax-pow 5229 ax-pr 5293 ax-un 7473 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2540 df-eu 2570 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ne 2935 df-ral 3058 df-rex 3059 df-rab 3062 df-v 3399 df-sbc 3680 df-dif 3844 df-un 3846 df-in 3848 df-ss 3858 df-symdif 4131 df-nul 4210 df-if 4412 df-pw 4487 df-sn 4514 df-pr 4516 df-op 4520 df-uni 4794 df-br 5028 df-opab 5090 df-mpt 5108 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 6291 df-fun 6335 df-fn 6336 df-f 6337 df-fo 6339 df-fv 6341 df-1st 7707 df-2nd 7708 df-txp 33786 df-bigcup 33790 |
This theorem is referenced by: fnbigcup 33833 |
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