Theorem iotass 5177

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 5160	. 2 |- ( iota x ph ) = U. { y | { x | ph } = { y } }
2		unieq 3805	. . . . . . . 8 |- ( { x | ph } = { y } -> U. { x | ph } = U. { y } )
3		vex 2733	. . . . . . . . 9 |- y e. _V
4	3	unisn 3812	. . . . . . . 8 |- U. { y } = y
5	2, 4	eqtrdi 2219	. . . . . . 7 |- ( { x | ph } = { y } -> U. { x | ph } = y )
6		df-pw 3568	. . . . . . . . . . 11 |- ~P A = { x | x C_ A }
7	6	sseq2i 3174	. . . . . . . . . 10 |- ( { x | ph } C_ ~P A <-> { x | ph } C_ { x | x C_ A } )
8		ss2ab 3215	. . . . . . . . . 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 3957	. . . . . . . 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 3170	. . . . . . . 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 3220	. . . 4 |- ( A. x ( ph -> x C_ A ) -> { y | { x | ph } = { y } } C_ { y | y C_ A } )
18		df-pw 3568	. . . 4 |- ~P A = { y | y C_ A }
19	17, 18	sseqtrrdi 3196	. . 3 |- ( A. x ( ph -> x C_ A ) -> { y | { x | ph } = { y } } C_ ~P A )
20		sspwuni 3957	. . 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 3193	1 |- ( A. x ( ph -> x C_ A ) -> ( iota x ph ) C_ A )

Syntax hints:   -> wi 4   A.wal 1346   = wceq 1348   {cab 2156   C_ wss 3121   ~Pcpw 3566   {csn 3583   U.cuni 3796   iotacio 5158

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-uni 3797  df-iota 5160

This theorem is referenced by:  fvss  5510  riotaexg  5813