Theorem fvss 5483

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 5179	. 2 |- ( F ` A ) = ( iota x A F x )
2		iotass 5153	. 2 |- ( A. x ( A F x -> x C_ B ) -> ( iota x A F x ) C_ B )
3	1, 2	eqsstrid 3174	1 |- ( A. x ( A F x -> x C_ B ) -> ( F ` A ) C_ B )

Syntax hints:   -> wi 4   A.wal 1333   C_ wss 3102   class class class wbr 3966   iotacio 5134   ` cfv 5171

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 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-uni 3774  df-iota 5136  df-fv 5179

This theorem is referenced by:  fvssunirng  5484  relfvssunirn  5485  sefvex  5490  fvmptss2  5544  tfrexlem  6282