Theorem iotass 5170

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	|- ( A. x ( ph -> x C_ A ) -> ( iota x ph ) C_ A )

Distinct variable group:   x, A

Allowed substitution hint:   ph( x)

Proof of Theorem iotass

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-iota 5153	. 2 |- ( iota x ph ) = U. { y | { x | ph } = { y } }
2		unieq 3798	. . . . . . . 8 |- ( { x | ph } = { y } -> U. { x | ph } = U. { y } )
3		vex 2729	. . . . . . . . 9 |- y e. _V
4	3	unisn 3805	. . . . . . . 8 |- U. { y } = y
5	2, 4	eqtrdi 2215	. . . . . . 7 |- ( { x | ph } = { y } -> U. { x | ph } = y )
6		df-pw 3561	. . . . . . . . . . 11 |- ~P A = { x | x C_ A }
7	6	sseq2i 3169	. . . . . . . . . 10 |- ( { x | ph } C_ ~P A <-> { x | ph } C_ { x | x C_ A } )
8		ss2ab 3210	. . . . . . . . . 10 |- ( { x | ph } C_ { x | x C_ A } <-> A. x ( ph -> x C_ A ) )
9	7, 8	bitri 183	. . . . . . . . 9 |- ( { x | ph } C_ ~P A <-> A. x ( ph -> x C_ A ) )
10	9	biimpri 132	. . . . . . . 8 |- ( A. x ( ph -> x C_ A ) -> { x | ph } C_ ~P A )
11		sspwuni 3950	. . . . . . . 8 |- ( { x | ph } C_ ~P A <-> U. { x | ph } C_ A )
12	10, 11	sylib 121	. . . . . . 7 |- ( A. x ( ph -> x C_ A ) -> U. { x | ph } C_ A )
13		sseq1 3165	. . . . . . . 8 |- ( U. { x | ph } = y -> ( U. { x | ph } C_ A <-> y C_ A ) )
14	13	biimpa 294	. . . . . . 7 |- ( ( U. { x | ph } = y /\ U. { x | ph } C_ A ) -> y C_ A )
15	5, 12, 14	syl2anr 288	. . . . . 6 |- ( ( A. x ( ph -> x C_ A ) /\ { x | ph } = { y } ) -> y C_ A )
16	15	ex 114	. . . . 5 |- ( A. x ( ph -> x C_ A ) -> ( { x | ph } = { y } -> y C_ A ) )
17	16	ss2abdv 3215	. . . 4 |- ( A. x ( ph -> x C_ A ) -> { y | { x | ph } = { y } } C_ { y | y C_ A } )
18		df-pw 3561	. . . 4 |- ~P A = { y | y C_ A }
19	17, 18	sseqtrrdi 3191	. . 3 |- ( A. x ( ph -> x C_ A ) -> { y | { x | ph } = { y } } C_ ~P A )
20		sspwuni 3950	. . 3 |- ( { y | { x | ph } = { y } } C_ ~P A <-> U. { y | { x | ph } = { y } } C_ A )
21	19, 20	sylib 121	. 2 |- ( A. x ( ph -> x C_ A ) -> U. { y | { x | ph } = { y } } C_ A )
22	1, 21	eqsstrid 3188	1 |- ( A. x ( ph -> x C_ A ) -> ( iota x ph ) C_ A )

Syntax hints:   -> wi 4   A.wal 1341   = wceq 1343   {cab 2151   C_ wss 3116   ~Pcpw 3559   {csn 3576   U.cuni 3789   iotacio 5151

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-uni 3790  df-iota 5153

This theorem is referenced by:  fvss  5500  riotaexg  5802