Theorem mptimass 5077

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 5076	. 2 |- ( ( x e. A |-> B ) " C ) = ran ( x e. ( A i^i C ) |-> B )
2		mptimass.1	. . . . 5 |- ( ph -> C C_ A )
3		sseqin2 3423	. . . . 5 |- ( C C_ A <-> ( A i^i C ) = C )
4	2, 3	sylib 122	. . . 4 |- ( ph -> ( A i^i C ) = C )
5	4	mpteq1d 4168	. . 3 |- ( ph -> ( x e. ( A i^i C ) |-> B ) = ( x e. C |-> B ) )
6	5	rneqd 4949	. 2 |- ( ph -> ran ( x e. ( A i^i C ) |-> B ) = ran ( x e. C |-> B ) )
7	1, 6	eqtrid 2274	1 |- ( ph -> ( ( x e. A |-> B ) " C ) = ran ( x e. C |-> B ) )

Syntax hints:   -> wi 4   = wceq 1395   i^i cin 3196   C_ wss 3197   |-> cmpt 4144   ran crn 4717   "cima 4719

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4083  df-opab 4145  df-mpt 4146  df-xp 4722  df-rel 4723  df-cnv 4724  df-dm 4726  df-rn 4727  df-res 4728  df-ima 4729

This theorem is referenced by: (None)