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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 3650 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 | |
2 | riotacl2 6524 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}) | |
3 | 1, 2 | sseldi 3566 | 1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1977 ∃!wreu 2898 {crab 2900 ℩crio 6510 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-un 3545 df-in 3547 df-ss 3554 df-sn 4126 df-pr 4128 df-uni 4373 df-iota 5768 df-riota 6511 |
This theorem is referenced by: riotaprop 6534 riotass2 6537 riotass 6538 riotaxfrd 6541 riotaclb 6548 supcl 8247 fisupcl 8258 htalem 8642 dfac8clem 8738 dfac2a 8835 fin23lem22 9032 zorn2lem1 9201 subcl 10159 divcl 10570 lbcl 10853 flcl 12458 cjf 13692 sqrtcl 13949 qnumdencl 15285 qnumdenbi 15290 catidcl 16166 lubcl 16808 glbcl 16821 ismgmid 17087 grpinvf 17289 pj1f 17933 mirf 25355 midf 25468 ismidb 25470 lmif 25477 islmib 25479 usgraidx2vlem1 25920 nbgraf1olem2 25971 frgrancvvdeqlem5 26561 grpoidcl 26752 grpoinvcl 26762 pjpreeq 27641 cnlnadjlem3 28312 adjbdln 28326 xdivcld 28962 cvmlift3lem3 30557 transportcl 31310 finxpreclem4 32407 poimirlem26 32605 iorlid 32827 riotaclbgBAD 33258 lshpkrlem2 33416 lshpkrcl 33421 cdleme25cl 34663 cdleme29cl 34683 cdlemefrs29clN 34705 cdlemk29-3 35217 cdlemkid5 35241 dihlsscpre 35541 mapdhcl 36034 hdmapcl 36140 hgmapcl 36199 wessf1ornlem 38366 fourierdlem50 39049 riotaeqimp 40338 uspgredg2vlem 40450 usgredg2vlem1 40452 frgrncvvdeqlem5 41473 |
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