Theorem fvss 5569

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 5263	. 2 |- ( F ` A ) = ( iota x A F x )
2		iotass 5233	. 2 |- ( A. x ( A F x -> x C_ B ) -> ( iota x A F x ) C_ B )
3	1, 2	eqsstrid 3226	1 |- ( A. x ( A F x -> x C_ B ) -> ( F ` A ) C_ B )

Syntax hints:   -> wi 4   A.wal 1362   C_ wss 3154   class class class wbr 4030   iotacio 5214   ` cfv 5255

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-uni 3837  df-iota 5216  df-fv 5263

This theorem is referenced by:  fvssunirng  5570  relfvssunirn  5571  sefvex  5576  fvmptss2  5633  tfrexlem  6389