Theorem mptima 5112

Description: Image of a function in maps-to notation. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

Assertion

Ref	Expression
mptima	|- ( ( x e. A |-> B ) " C ) = ran ( x e. ( A i^i C ) |-> B )

Distinct variable groups:   x, A   x, C

Allowed substitution hint:   B( x)

Proof of Theorem mptima

Step	Hyp	Ref	Expression
1		df-ima 4761	. 2 |- ( ( x e. A |-> B ) " C ) = ran ( ( x e. A |-> B ) |` C )
2		resmpt3 5086	. . 3 |- ( ( x e. A |-> B ) |` C ) = ( x e. ( A i^i C ) |-> B )
3	2	rneqi 4984	. 2 |- ran ( ( x e. A |-> B ) |` C ) = ran ( x e. ( A i^i C ) |-> B )
4	1, 3	eqtri 2253	1 |- ( ( x e. A |-> B ) " C ) = ran ( x e. ( A i^i C ) |-> B )

Syntax hints:   = wceq 1398   i^i cin 3209   |-> cmpt 4170   ran crn 4749   |` cres 4750   "cima 4751

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-14 2206  ax-ext 2214  ax-sep 4227  ax-pow 4286  ax-pr 4321

This theorem depends on definitions:  df-bi 117  df-3an 1007  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 2814  df-un 3214  df-in 3216  df-ss 3223  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-br 4109  df-opab 4171  df-mpt 4172  df-xp 4754  df-rel 4755  df-cnv 4756  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761

This theorem is referenced by:  mptimass  5113  elply2  15587