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Mirrors > Home > MPE Home > Th. List > riotacl | Structured version Visualization version GIF version |
Description: Closure of restricted iota. (Contributed by NM, 21-Aug-2011.) |
Ref | Expression |
---|---|
riotacl | ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssrab2 4077 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 | |
2 | riotacl2 7385 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}) | |
3 | 1, 2 | sselid 3980 | 1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ∃!wreu 3373 {crab 3431 ℩crio 7367 |
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-12 2170 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-tru 1543 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-reu 3376 df-rab 3432 df-v 3475 df-sbc 3778 df-un 3953 df-in 3955 df-ss 3965 df-sn 4629 df-pr 4631 df-uni 4909 df-iota 6495 df-riota 7368 |
This theorem is referenced by: riotaeqimp 7395 riotaprop 7396 riotass2 7399 riotass 7400 riotaxfrd 7403 riotaclb 7410 supcl 9459 fisupcl 9470 ttrcltr 9717 htalem 9897 dfac8clem 10033 dfac2a 10130 fin23lem22 10328 zorn2lem1 10497 subcl 11466 divcl 11885 lbcl 12172 flcl 13767 cjf 15058 sqrtcl 15315 qnumdencl 16682 qnumdenbi 16687 catidcl 17633 lubcl 18320 glbcl 18333 ismgmid 18596 grpinvfval 18906 grpinvf 18914 pj1f 19613 nosupno 27549 nosupbday 27551 nosupbnd1 27560 noinfno 27564 noinfbday 27566 noinfbnd1 27575 scutcut 27647 divsclw 28007 mirf 28344 midf 28460 ismidb 28462 lmif 28469 islmib 28471 uspgredg2vlem 28913 usgredg2vlem1 28915 frgrncvvdeqlem4 29988 grpoidcl 30200 grpoinvcl 30210 pjpreeq 31084 cnlnadjlem3 31755 adjbdln 31769 xdivcld 32522 cvmlift3lem3 34776 transportcl 35475 finxpreclem4 36739 poimirlem26 36978 iorlid 37190 riotaclbgBAD 38288 lshpkrlem2 38445 lshpkrcl 38450 cdleme25cl 39692 cdleme29cl 39712 cdlemefrs29clN 39734 cdlemk29-3 40246 cdlemkid5 40270 dihlsscpre 40569 mapdhcl 41062 hdmapcl 41165 hgmapcl 41224 fsuppind 41625 rernegcl 41707 rersubcl 41714 sn-subcl 41763 tfsconcatfv 42554 wessf1ornlem 44343 fourierdlem50 45331 |
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