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Theorem iotass 5073
 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 5056 . 2
2 unieq 3713 . . . . . . . 8
3 vex 2661 . . . . . . . . 9
43unisn 3720 . . . . . . . 8
52, 4syl6eq 2164 . . . . . . 7
6 df-pw 3480 . . . . . . . . . . 11
76sseq2i 3092 . . . . . . . . . 10
8 ss2ab 3133 . . . . . . . . . 10
97, 8bitri 183 . . . . . . . . 9
109biimpri 132 . . . . . . . 8
11 sspwuni 3865 . . . . . . . 8
1210, 11sylib 121 . . . . . . 7
13 sseq1 3088 . . . . . . . 8
1413biimpa 292 . . . . . . 7
155, 12, 14syl2anr 286 . . . . . 6
1615ex 114 . . . . 5
1716ss2abdv 3138 . . . 4
18 df-pw 3480 . . . 4
1917, 18sseqtrrdi 3114 . . 3
20 sspwuni 3865 . . 3
2119, 20sylib 121 . 2
221, 21eqsstrid 3111 1
 Colors of variables: wff set class Syntax hints:   wi 4  wal 1312   wceq 1314  cab 2101   wss 3039  cpw 3478  csn 3495  cuni 3704  cio 5054 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 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097 This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-uni 3705  df-iota 5056 This theorem is referenced by:  fvss  5401  riotaexg  5700
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