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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 :VāontoāV |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uniexg 7734 | . . . 4 ⢠(š„ ā V ā āŖ š„ ā V) | |
2 | 1 | rgen 3062 | . . 3 ā¢ āš„ ā V āŖ š„ ā V |
3 | dfbigcup2 35176 | . . . 4 ⢠Bigcup = (š„ ā V ⦠⪠š„) | |
4 | 3 | mptfng 6689 | . . 3 ⢠(āš„ ā V āŖ š„ ā V ā Bigcup Fn V) |
5 | 2, 4 | mpbi 229 | . 2 ⢠Bigcup Fn V |
6 | 3 | rnmpt 5954 | . . 3 ⢠ran Bigcup = {š¦ ā£ āš„ ā V š¦ = āŖ š„} |
7 | vex 3477 | . . . . 5 ⢠š¦ ā V | |
8 | vsnex 5429 | . . . . . 6 ⢠{š¦} ā V | |
9 | unisnv 4931 | . . . . . . 7 ⢠⪠{š¦} = š¦ | |
10 | 9 | eqcomi 2740 | . . . . . 6 ⢠š¦ = āŖ {š¦} |
11 | unieq 4919 | . . . . . . 7 ⢠(š„ = {š¦} ā āŖ š„ = āŖ {š¦}) | |
12 | 11 | rspceeqv 3633 | . . . . . 6 ⢠(({š¦} ā V ā§ š¦ = āŖ {š¦}) ā āš„ ā V š¦ = āŖ š„) |
13 | 8, 10, 12 | mp2an 689 | . . . . 5 ā¢ āš„ ā V š¦ = āŖ š„ |
14 | 7, 13 | 2th 264 | . . . 4 ⢠(š¦ ā V ā āš„ ā V š¦ = āŖ š„) |
15 | 14 | eqabi 2868 | . . 3 ⢠V = {š¦ ā£ āš„ ā V š¦ = āŖ š„} |
16 | 6, 15 | eqtr4i 2762 | . 2 ⢠ran Bigcup = V |
17 | df-fo 6549 | . 2 ⢠( Bigcup :VāontoāV ā ( Bigcup Fn V ā§ ran Bigcup = V)) | |
18 | 5, 16, 17 | mpbir2an 708 | 1 ⢠Bigcup :VāontoāV |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1540 ā wcel 2105 {cab 2708 āwral 3060 āwrex 3069 Vcvv 3473 {csn 4628 āŖ cuni 4908 ran crn 5677 Fn wfn 6538 āontoāwfo 6541 Bigcup cbigcup 35111 |
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 1912 ax-6 1970 ax-7 2010 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2153 ax-12 2170 ax-ext 2702 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7729 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1781 df-nf 1785 df-sb 2067 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-symdif 4242 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-eprel 5580 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-fo 6549 df-fv 6551 df-1st 7979 df-2nd 7980 df-txp 35131 df-bigcup 35135 |
This theorem is referenced by: fnbigcup 35178 |
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