Theorem mptimass 5040

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 5039	. 2 |- ( ( x e. A |-> B ) " C ) = ran ( x e. ( A i^i C ) |-> B )
2		mptimass.1	. . . . 5 |- ( ph -> C C_ A )
3		sseqin2 3393	. . . . 5 |- ( C C_ A <-> ( A i^i C ) = C )
4	2, 3	sylib 122	. . . 4 |- ( ph -> ( A i^i C ) = C )
5	4	mpteq1d 4133	. . 3 |- ( ph -> ( x e. ( A i^i C ) |-> B ) = ( x e. C |-> B ) )
6	5	rneqd 4912	. 2 |- ( ph -> ran ( x e. ( A i^i C ) |-> B ) = ran ( x e. C |-> B ) )
7	1, 6	eqtrid 2251	1 |- ( ph -> ( ( x e. A |-> B ) " C ) = ran ( x e. C |-> B ) )

Syntax hints:   -> wi 4   = wceq 1373   i^i cin 3166   C_ wss 3167   |-> cmpt 4109   ran crn 4680   "cima 4682

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-14 2180  ax-ext 2188  ax-sep 4166  ax-pow 4222  ax-pr 4257

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-un 3171  df-in 3173  df-ss 3180  df-pw 3619  df-sn 3640  df-pr 3641  df-op 3643  df-br 4048  df-opab 4110  df-mpt 4111  df-xp 4685  df-rel 4686  df-cnv 4687  df-dm 4689  df-rn 4690  df-res 4691  df-ima 4692

This theorem is referenced by: (None)