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Mirrors > Home > ILE Home > Th. List > riotacl | 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 3177 | . 2 ⊢ {𝑥 ∈ 𝐴 ∣ 𝜑} ⊆ 𝐴 | |
2 | riotacl2 5736 | . 2 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ {𝑥 ∈ 𝐴 ∣ 𝜑}) | |
3 | 1, 2 | sseldi 3090 | 1 ⊢ (∃!𝑥 ∈ 𝐴 𝜑 → (℩𝑥 ∈ 𝐴 𝜑) ∈ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1480 ∃!wreu 2416 {crab 2418 ℩crio 5722 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2000 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-rex 2420 df-reu 2421 df-rab 2423 df-v 2683 df-sbc 2905 df-un 3070 df-in 3072 df-ss 3079 df-sn 3528 df-pr 3529 df-uni 3732 df-iota 5083 df-riota 5723 |
This theorem is referenced by: riotaprop 5746 riotass2 5749 riotass 5750 acexmidlemcase 5762 supclti 6878 caucvgsrlemcl 7590 caucvgsrlemgt1 7596 axcaucvglemcl 7696 subval 7947 subcl 7954 divvalap 8427 divclap 8431 lbcl 8697 divfnzn 9406 flqcl 10039 flapcl 10041 cjval 10610 cjth 10611 cjf 10612 oddpwdclemodd 11839 oddpwdclemdc 11840 oddpwdc 11841 qnumdencl 11854 qnumdenbi 11859 |
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