Theorem resmpt3 4933

Description: Unconditional restriction of the mapping operation. (Contributed by Stefan O'Rear, 24-Jan-2015.) (Proof shortened by Mario Carneiro, 22-Mar-2015.)

Assertion

Ref	Expression
resmpt3	|- ( ( x e. A |-> C ) |` B ) = ( x e. ( A i^i B ) |-> C )

Distinct variable groups:   x, A   x, B

Allowed substitution hint:   C( x)

Proof of Theorem resmpt3

Step	Hyp	Ref	Expression
1		resres 4896	. 2 |- ( ( ( x e. A |-> C ) |` A ) |` B ) = ( ( x e. A |-> C ) |` ( A i^i B ) )
2		ssid 3162	. . . 4 |- A C_ A
3		resmpt 4932	. . . 4 |- ( A C_ A -> ( ( x e. A |-> C ) |` A ) = ( x e. A |-> C ) )
4	2, 3	ax-mp 5	. . 3 |- ( ( x e. A |-> C ) |` A ) = ( x e. A |-> C )
5	4	reseq1i 4880	. 2 |- ( ( ( x e. A |-> C ) |` A ) |` B ) = ( ( x e. A |-> C ) |` B )
6		inss1 3342	. . 3 |- ( A i^i B ) C_ A
7		resmpt 4932	. . 3 |- ( ( A i^i B ) C_ A -> ( ( x e. A |-> C ) |` ( A i^i B ) ) = ( x e. ( A i^i B ) |-> C ) )
8	6, 7	ax-mp 5	. 2 |- ( ( x e. A |-> C ) |` ( A i^i B ) ) = ( x e. ( A i^i B ) |-> C )
9	1, 5, 8	3eqtr3i 2194	1 |- ( ( x e. A |-> C ) |` B ) = ( x e. ( A i^i B ) |-> C )

Syntax hints:   = wceq 1343   i^i cin 3115   C_ wss 3116   |-> cmpt 4043   |` cres 4606

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-opab 4044  df-mpt 4045  df-xp 4610  df-rel 4611  df-res 4616

This theorem is referenced by:  offres  6103