Theorem iotass 4904

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 4887	. 2 |- ( iota x ph ) = U. { y | { x | ph } = { y } }
2		unieq 3610	. . . . . . . 8 |- ( { x | ph } = { y } -> U. { x | ph } = U. { y } )
3		vex 2604	. . . . . . . . 9 |- y e. _V
4	3	unisn 3617	. . . . . . . 8 |- U. { y } = y
5	2, 4	syl6eq 2129	. . . . . . 7 |- ( { x | ph } = { y } -> U. { x | ph } = y )
6		df-pw 3384	. . . . . . . . . . 11 |- ~P A = { x | x C_ A }
7	6	sseq2i 3024	. . . . . . . . . 10 |- ( { x | ph } C_ ~P A <-> { x | ph } C_ { x | x C_ A } )
8		ss2ab 3062	. . . . . . . . . 10 |- ( { x | ph } C_ { x | x C_ A } <-> A. x ( ph -> x C_ A ) )
9	7, 8	bitri 182	. . . . . . . . 9 |- ( { x | ph } C_ ~P A <-> A. x ( ph -> x C_ A ) )
10	9	biimpri 131	. . . . . . . 8 |- ( A. x ( ph -> x C_ A ) -> { x | ph } C_ ~P A )
11		sspwuni 3760	. . . . . . . 8 |- ( { x | ph } C_ ~P A <-> U. { x | ph } C_ A )
12	10, 11	sylib 120	. . . . . . 7 |- ( A. x ( ph -> x C_ A ) -> U. { x | ph } C_ A )
13		sseq1 3020	. . . . . . . 8 |- ( U. { x | ph } = y -> ( U. { x | ph } C_ A <-> y C_ A ) )
14	13	biimpa 290	. . . . . . 7 |- ( ( U. { x | ph } = y /\ U. { x | ph } C_ A ) -> y C_ A )
15	5, 12, 14	syl2anr 284	. . . . . 6 |- ( ( A. x ( ph -> x C_ A ) /\ { x | ph } = { y } ) -> y C_ A )
16	15	ex 113	. . . . 5 |- ( A. x ( ph -> x C_ A ) -> ( { x | ph } = { y } -> y C_ A ) )
17	16	ss2abdv 3067	. . . 4 |- ( A. x ( ph -> x C_ A ) -> { y | { x | ph } = { y } } C_ { y | y C_ A } )
18		df-pw 3384	. . . 4 |- ~P A = { y | y C_ A }
19	17, 18	syl6sseqr 3046	. . 3 |- ( A. x ( ph -> x C_ A ) -> { y | { x | ph } = { y } } C_ ~P A )
20		sspwuni 3760	. . 3 |- ( { y | { x | ph } = { y } } C_ ~P A <-> U. { y | { x | ph } = { y } } C_ A )
21	19, 20	sylib 120	. 2 |- ( A. x ( ph -> x C_ A ) -> U. { y | { x | ph } = { y } } C_ A )
22	1, 21	syl5eqss 3043	1 |- ( A. x ( ph -> x C_ A ) -> ( iota x ph ) C_ A )

Syntax hints:   -> wi 4   A.wal 1282   = wceq 1284   {cab 2067   C_ wss 2973   ~Pcpw 3382   {csn 3398   U.cuni 3601   iotacio 4885

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-uni 3602  df-iota 4887

This theorem is referenced by:  fvss  5209  riotaexg  5492