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Theorem iotass 4912
 Description: Value of iota based on a proposition which holds only for values which are subsets of a given class. (Contributed by Mario Carneiro and Jim Kingdon, 21-Dec-2018.)
Assertion
Ref Expression
iotass
Distinct variable group:   ,
Allowed substitution hint:   ()

Proof of Theorem iotass
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 df-iota 4895 . 2
2 unieq 3617 . . . . . . . 8
3 vex 2577 . . . . . . . . 9
43unisn 3624 . . . . . . . 8
52, 4syl6eq 2104 . . . . . . 7
6 df-pw 3389 . . . . . . . . . . 11
76sseq2i 2998 . . . . . . . . . 10
8 ss2ab 3036 . . . . . . . . . 10
97, 8bitri 177 . . . . . . . . 9
109biimpri 128 . . . . . . . 8
11 sspwuni 3767 . . . . . . . 8
1210, 11sylib 131 . . . . . . 7
13 sseq1 2994 . . . . . . . 8
1413biimpa 284 . . . . . . 7
155, 12, 14syl2anr 278 . . . . . 6
1615ex 112 . . . . 5
1716ss2abdv 3041 . . . 4
18 df-pw 3389 . . . 4
1917, 18syl6sseqr 3020 . . 3
20 sspwuni 3767 . . 3
2119, 20sylib 131 . 2
221, 21syl5eqss 3017 1
 Colors of variables: wff set class Syntax hints:   wi 4  wal 1257   wceq 1259  cab 2042   wss 2945  cpw 3387  csn 3403  cuni 3608  cio 4893 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-uni 3609  df-iota 4895 This theorem is referenced by:  fvss  5217  riotaexg  5500
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