Theorem mptimass 5018

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

Hypothesis

Ref	Expression
mptimass.1	|- ( ph -> C C_ A )

Assertion

Ref	Expression
mptimass	|- ( ph -> ( ( x e. A |-> B ) " C ) = ran ( x e. C |-> B ) )

Distinct variable groups:   x, A   x, C

Allowed substitution hints:   ph( x)   B( x)

Proof of Theorem mptimass

Step	Hyp	Ref	Expression
1		mptima 5017	. 2 |- ( ( x e. A |-> B ) " C ) = ran ( x e. ( A i^i C ) |-> B )
2		mptimass.1	. . . . 5 |- ( ph -> C C_ A )
3		sseqin2 3378	. . . . 5 |- ( C C_ A <-> ( A i^i C ) = C )
4	2, 3	sylib 122	. . . 4 |- ( ph -> ( A i^i C ) = C )
5	4	mpteq1d 4114	. . 3 |- ( ph -> ( x e. ( A i^i C ) |-> B ) = ( x e. C |-> B ) )
6	5	rneqd 4891	. 2 |- ( ph -> ran ( x e. ( A i^i C ) |-> B ) = ran ( x e. C |-> B ) )
7	1, 6	eqtrid 2238	1 |- ( ph -> ( ( x e. A |-> B ) " C ) = ran ( x e. C |-> B ) )

Syntax hints:   -> wi 4   = wceq 1364   i^i cin 3152   C_ wss 3153   |-> cmpt 4090   ran crn 4660   "cima 4662

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-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  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 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-mpt 4092  df-xp 4665  df-rel 4666  df-cnv 4667  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672

This theorem is referenced by: (None)