![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 3908 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 | |
2 | riotacl2 6896 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}) | |
3 | 1, 2 | sseldi 3819 | 1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 ∃!wreu 3092 {crab 3094 ℩crio 6882 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-un 3797 df-in 3799 df-ss 3806 df-sn 4399 df-pr 4401 df-uni 4672 df-iota 6099 df-riota 6883 |
This theorem is referenced by: riotaeqimp 6906 riotaprop 6907 riotass2 6910 riotass 6911 riotaxfrd 6914 riotaclb 6921 supcl 8652 fisupcl 8663 htalem 9056 dfac8clem 9188 dfac2a 9285 fin23lem22 9484 zorn2lem1 9653 subcl 10621 divcl 11039 lbcl 11328 flcl 12915 cjf 14251 sqrtcl 14508 qnumdencl 15851 qnumdenbi 15856 catidcl 16728 lubcl 17371 glbcl 17384 ismgmid 17650 grpinvf 17853 pj1f 18494 mirf 26011 midf 26124 ismidb 26126 lmif 26133 islmib 26135 uspgredg2vlem 26569 usgredg2vlem1 26571 frgrncvvdeqlem4 27710 grpoidcl 27941 grpoinvcl 27951 pjpreeq 28829 cnlnadjlem3 29500 adjbdln 29514 xdivcld 30193 cvmlift3lem3 31902 nosupno 32438 nosupbday 32440 nosupbnd1 32449 scutcut 32501 transportcl 32729 finxpreclem4 33826 poimirlem26 34063 iorlid 34283 riotaclbgBAD 35110 lshpkrlem2 35267 lshpkrcl 35272 cdleme25cl 36513 cdleme29cl 36533 cdlemefrs29clN 36555 cdlemk29-3 37067 cdlemkid5 37091 dihlsscpre 37390 mapdhcl 37883 hdmapcl 37986 hgmapcl 38045 rernegcl 38181 rersubcl 38189 wessf1ornlem 40298 fourierdlem50 41304 |
Copyright terms: Public domain | W3C validator |