Theorem fvss 5443

Description: The value of a function is a subset of B if every element that could be a candidate for the value is a subset of B. (Contributed by Mario Carneiro, 24-May-2019.)

Assertion

Ref	Expression
fvss	|- ( A. x ( A F x -> x C_ B ) -> ( F ` A ) C_ B )

Distinct variable groups:   x, A   x, B   x, F

Proof of Theorem fvss

Step	Hyp	Ref	Expression
1		df-fv 5139	. 2 |- ( F ` A ) = ( iota x A F x )
2		iotass 5113	. 2 |- ( A. x ( A F x -> x C_ B ) -> ( iota x A F x ) C_ B )
3	1, 2	eqsstrid 3148	1 |- ( A. x ( A F x -> x C_ B ) -> ( F ` A ) C_ B )

Syntax hints:   -> wi 4   A.wal 1330   C_ wss 3076   class class class wbr 3937   iotacio 5094   ` cfv 5131

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-uni 3745  df-iota 5096  df-fv 5139

This theorem is referenced by:  fvssunirng  5444  relfvssunirn  5445  sefvex  5450  fvmptss2  5504  tfrexlem  6239