Theorem mptimass 5095

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 5094	. 2 |- ( ( x e. A |-> B ) " C ) = ran ( x e. ( A i^i C ) |-> B )
2		mptimass.1	. . . . 5 |- ( ph -> C C_ A )
3		sseqin2 3428	. . . . 5 |- ( C C_ A <-> ( A i^i C ) = C )
4	2, 3	sylib 122	. . . 4 |- ( ph -> ( A i^i C ) = C )
5	4	mpteq1d 4179	. . 3 |- ( ph -> ( x e. ( A i^i C ) |-> B ) = ( x e. C |-> B ) )
6	5	rneqd 4967	. 2 |- ( ph -> ran ( x e. ( A i^i C ) |-> B ) = ran ( x e. C |-> B ) )
7	1, 6	eqtrid 2276	1 |- ( ph -> ( ( x e. A |-> B ) " C ) = ran ( x e. C |-> B ) )

Syntax hints:   -> wi 4   = wceq 1398   i^i cin 3200   C_ wss 3201   |-> cmpt 4155   ran crn 4732   "cima 4734

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 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-mpt 4157  df-xp 4737  df-rel 4738  df-cnv 4739  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744

This theorem is referenced by: (None)