Theorem fvss 5649

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 5332	. 2 |- ( F ` A ) = ( iota x A F x )
2		iotass 5302	. 2 |- ( A. x ( A F x -> x C_ B ) -> ( iota x A F x ) C_ B )
3	1, 2	eqsstrid 3271	1 |- ( A. x ( A F x -> x C_ B ) -> ( F ` A ) C_ B )

Syntax hints:   -> wi 4   A.wal 1393   C_ wss 3198   class class class wbr 4086   iotacio 5282   ` cfv 5324

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-uni 3892  df-iota 5284  df-fv 5332

This theorem is referenced by:  fvssunirng  5650  relfvssunirn  5651  sefvex  5656  fvmptss2  5717  tfrexlem  6495