Theorem mptima 5056

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 4709	. 2 |- ( ( x e. A |-> B ) " C ) = ran ( ( x e. A |-> B ) |` C )
2		resmpt3 5030	. . 3 |- ( ( x e. A |-> B ) |` C ) = ( x e. ( A i^i C ) |-> B )
3	2	rneqi 4928	. 2 |- ran ( ( x e. A |-> B ) |` C ) = ran ( x e. ( A i^i C ) |-> B )
4	1, 3	eqtri 2230	1 |- ( ( x e. A |-> B ) " C ) = ran ( x e. ( A i^i C ) |-> B )

Syntax hints:   = wceq 1375   i^i cin 3176   |-> cmpt 4124   ran crn 4697   |` cres 4698   "cima 4699

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-pow 4237  ax-pr 4272

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ral 2493  df-rex 2494  df-v 2781  df-un 3181  df-in 3183  df-ss 3190  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-br 4063  df-opab 4125  df-mpt 4126  df-xp 4702  df-rel 4703  df-cnv 4704  df-dm 4706  df-rn 4707  df-res 4708  df-ima 4709

This theorem is referenced by:  mptimass  5057  elply2  15374