Theorem fvss 5684

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 5360	. 2 |- ( F ` A ) = ( iota x A F x )
2		iotass 5330	. 2 |- ( A. x ( A F x -> x C_ B ) -> ( iota x A F x ) C_ B )
3	1, 2	eqsstrid 3284	1 |- ( A. x ( A F x -> x C_ B ) -> ( F ` A ) C_ B )

Syntax hints:   -> wi 4   A.wal 1396   C_ wss 3211   class class class wbr 4109   iotacio 5310   ` cfv 5352

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-uni 3915  df-iota 5312  df-fv 5360

This theorem is referenced by:  fvssunirng  5685  relfvssunirn  5686  sefvex  5691  fvmptss2  5752  tfrexlem  6565