Theorem iotass 5210

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 5193	. 2 |- ( iota x ph ) = U. { y | { x | ph } = { y } }
2		unieq 3833	. . . . . . . 8 |- ( { x | ph } = { y } -> U. { x | ph } = U. { y } )
3		vex 2755	. . . . . . . . 9 |- y e. _V
4	3	unisn 3840	. . . . . . . 8 |- U. { y } = y
5	2, 4	eqtrdi 2238	. . . . . . 7 |- ( { x | ph } = { y } -> U. { x | ph } = y )
6		df-pw 3592	. . . . . . . . . . 11 |- ~P A = { x | x C_ A }
7	6	sseq2i 3197	. . . . . . . . . 10 |- ( { x | ph } C_ ~P A <-> { x | ph } C_ { x | x C_ A } )
8		ss2ab 3238	. . . . . . . . . 10 |- ( { x | ph } C_ { x | x C_ A } <-> A. x ( ph -> x C_ A ) )
9	7, 8	bitri 184	. . . . . . . . 9 |- ( { x | ph } C_ ~P A <-> A. x ( ph -> x C_ A ) )
10	9	biimpri 133	. . . . . . . 8 |- ( A. x ( ph -> x C_ A ) -> { x | ph } C_ ~P A )
11		sspwuni 3986	. . . . . . . 8 |- ( { x | ph } C_ ~P A <-> U. { x | ph } C_ A )
12	10, 11	sylib 122	. . . . . . 7 |- ( A. x ( ph -> x C_ A ) -> U. { x | ph } C_ A )
13		sseq1 3193	. . . . . . . 8 |- ( U. { x | ph } = y -> ( U. { x | ph } C_ A <-> y C_ A ) )
14	13	biimpa 296	. . . . . . 7 |- ( ( U. { x | ph } = y /\ U. { x | ph } C_ A ) -> y C_ A )
15	5, 12, 14	syl2anr 290	. . . . . 6 |- ( ( A. x ( ph -> x C_ A ) /\ { x | ph } = { y } ) -> y C_ A )
16	15	ex 115	. . . . 5 |- ( A. x ( ph -> x C_ A ) -> ( { x | ph } = { y } -> y C_ A ) )
17	16	ss2abdv 3243	. . . 4 |- ( A. x ( ph -> x C_ A ) -> { y | { x | ph } = { y } } C_ { y | y C_ A } )
18		df-pw 3592	. . . 4 |- ~P A = { y | y C_ A }
19	17, 18	sseqtrrdi 3219	. . 3 |- ( A. x ( ph -> x C_ A ) -> { y | { x | ph } = { y } } C_ ~P A )
20		sspwuni 3986	. . 3 |- ( { y | { x | ph } = { y } } C_ ~P A <-> U. { y | { x | ph } = { y } } C_ A )
21	19, 20	sylib 122	. 2 |- ( A. x ( ph -> x C_ A ) -> U. { y | { x | ph } = { y } } C_ A )
22	1, 21	eqsstrid 3216	1 |- ( A. x ( ph -> x C_ A ) -> ( iota x ph ) C_ A )

Syntax hints:   -> wi 4   A.wal 1362   = wceq 1364   {cab 2175   C_ wss 3144   ~Pcpw 3590   {csn 3607   U.cuni 3824   iotacio 5191

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-uni 3825  df-iota 5193

This theorem is referenced by:  fvss  5544  riotaexg  5851