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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 7593 | . . . 4 ⊢ (𝑥 ∈ V → ∪ 𝑥 ∈ V) | |
2 | 1 | rgen 3074 | . . 3 ⊢ ∀𝑥 ∈ V ∪ 𝑥 ∈ V |
3 | dfbigcup2 34201 | . . . 4 ⊢ Bigcup = (𝑥 ∈ V ↦ ∪ 𝑥) | |
4 | 3 | mptfng 6572 | . . 3 ⊢ (∀𝑥 ∈ V ∪ 𝑥 ∈ V ↔ Bigcup Fn V) |
5 | 2, 4 | mpbi 229 | . 2 ⊢ Bigcup Fn V |
6 | 3 | rnmpt 5864 | . . 3 ⊢ ran Bigcup = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ 𝑥} |
7 | vex 3436 | . . . . 5 ⊢ 𝑦 ∈ V | |
8 | snex 5354 | . . . . . 6 ⊢ {𝑦} ∈ V | |
9 | 7 | unisn 4861 | . . . . . . 7 ⊢ ∪ {𝑦} = 𝑦 |
10 | 9 | eqcomi 2747 | . . . . . 6 ⊢ 𝑦 = ∪ {𝑦} |
11 | unieq 4850 | . . . . . . 7 ⊢ (𝑥 = {𝑦} → ∪ 𝑥 = ∪ {𝑦}) | |
12 | 11 | rspceeqv 3575 | . . . . . 6 ⊢ (({𝑦} ∈ V ∧ 𝑦 = ∪ {𝑦}) → ∃𝑥 ∈ V 𝑦 = ∪ 𝑥) |
13 | 8, 10, 12 | mp2an 689 | . . . . 5 ⊢ ∃𝑥 ∈ V 𝑦 = ∪ 𝑥 |
14 | 7, 13 | 2th 263 | . . . 4 ⊢ (𝑦 ∈ V ↔ ∃𝑥 ∈ V 𝑦 = ∪ 𝑥) |
15 | 14 | abbi2i 2879 | . . 3 ⊢ V = {𝑦 ∣ ∃𝑥 ∈ V 𝑦 = ∪ 𝑥} |
16 | 6, 15 | eqtr4i 2769 | . 2 ⊢ ran Bigcup = V |
17 | df-fo 6439 | . 2 ⊢ ( Bigcup :V–onto→V ↔ ( Bigcup Fn V ∧ ran Bigcup = V)) | |
18 | 5, 16, 17 | mpbir2an 708 | 1 ⊢ Bigcup :V–onto→V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1539 ∈ wcel 2106 {cab 2715 ∀wral 3064 ∃wrex 3065 Vcvv 3432 {csn 4561 ∪ cuni 4839 ran crn 5590 Fn wfn 6428 –onto→wfo 6431 Bigcup cbigcup 34136 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-symdif 4176 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-eprel 5495 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-fo 6439 df-fv 6441 df-1st 7831 df-2nd 7832 df-txp 34156 df-bigcup 34160 |
This theorem is referenced by: fnbigcup 34203 |
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