Theorem iotass 4884

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 4867	. 2 |- ( iota x ph ) = U. { y | { x | ph } = { y } }
2		unieq 3589	. . . . . . . 8 |- ( { x | ph } = { y } -> U. { x | ph } = U. { y } )
3		vex 2560	. . . . . . . . 9 |- y e. _V
4	3	unisn 3596	. . . . . . . 8 |- U. { y } = y
5	2, 4	syl6eq 2088	. . . . . . 7 |- ( { x | ph } = { y } -> U. { x | ph } = y )
6		df-pw 3361	. . . . . . . . . . 11 |- ~P A = { x | x C_ A }
7	6	sseq2i 2970	. . . . . . . . . 10 |- ( { x | ph } C_ ~P A <-> { x | ph } C_ { x | x C_ A } )
8		ss2ab 3008	. . . . . . . . . 10 |- ( { x | ph } C_ { x | x C_ A } <-> A. x ( ph -> x C_ A ) )
9	7, 8	bitri 173	. . . . . . . . 9 |- ( { x | ph } C_ ~P A <-> A. x ( ph -> x C_ A ) )
10	9	biimpri 124	. . . . . . . 8 |- ( A. x ( ph -> x C_ A ) -> { x | ph } C_ ~P A )
11		sspwuni 3739	. . . . . . . 8 |- ( { x | ph } C_ ~P A <-> U. { x | ph } C_ A )
12	10, 11	sylib 127	. . . . . . 7 |- ( A. x ( ph -> x C_ A ) -> U. { x | ph } C_ A )
13		sseq1 2966	. . . . . . . 8 |- ( U. { x | ph } = y -> ( U. { x | ph } C_ A <-> y C_ A ) )
14	13	biimpa 280	. . . . . . 7 |- ( ( U. { x | ph } = y /\ U. { x | ph } C_ A ) -> y C_ A )
15	5, 12, 14	syl2anr 274	. . . . . 6 |- ( ( A. x ( ph -> x C_ A ) /\ { x | ph } = { y } ) -> y C_ A )
16	15	ex 108	. . . . 5 |- ( A. x ( ph -> x C_ A ) -> ( { x | ph } = { y } -> y C_ A ) )
17	16	ss2abdv 3013	. . . 4 |- ( A. x ( ph -> x C_ A ) -> { y | { x | ph } = { y } } C_ { y | y C_ A } )
18		df-pw 3361	. . . 4 |- ~P A = { y | y C_ A }
19	17, 18	syl6sseqr 2992	. . 3 |- ( A. x ( ph -> x C_ A ) -> { y | { x | ph } = { y } } C_ ~P A )
20		sspwuni 3739	. . 3 |- ( { y | { x | ph } = { y } } C_ ~P A <-> U. { y | { x | ph } = { y } } C_ A )
21	19, 20	sylib 127	. 2 |- ( A. x ( ph -> x C_ A ) -> U. { y | { x | ph } = { y } } C_ A )
22	1, 21	syl5eqss 2989	1 |- ( A. x ( ph -> x C_ A ) -> ( iota x ph ) C_ A )

Syntax hints:   -> wi 4   A.wal 1241   = wceq 1243   {cab 2026   C_ wss 2917   ~Pcpw 3359   {csn 3375   U.cuni 3580   iotacio 4865

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-uni 3581  df-iota 4867

This theorem is referenced by:  fvss  5189  riotaexg  5472