Theorem resmpt3 4991

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 4954	. 2 |- ( ( ( x e. A |-> C ) |` A ) |` B ) = ( ( x e. A |-> C ) |` ( A i^i B ) )
2		ssid 3199	. . . 4 |- A C_ A
3		resmpt 4990	. . . 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 4938	. 2 |- ( ( ( x e. A |-> C ) |` A ) |` B ) = ( ( x e. A |-> C ) |` B )
6		inss1 3379	. . 3 |- ( A i^i B ) C_ A
7		resmpt 4990	. . 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 2222	1 |- ( ( x e. A |-> C ) |` B ) = ( x e. ( A i^i B ) |-> C )

Syntax hints:   = wceq 1364   i^i cin 3152   C_ wss 3153   |-> cmpt 4090   |` cres 4661

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-opab 4091  df-mpt 4092  df-xp 4665  df-rel 4666  df-res 4671

This theorem is referenced by:  mptima  5017  offres  6187