Theorem resmpt3 4957

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 4920	. 2 |- ( ( ( x e. A |-> C ) |` A ) |` B ) = ( ( x e. A |-> C ) |` ( A i^i B ) )
2		ssid 3176	. . . 4 |- A C_ A
3		resmpt 4956	. . . 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 4904	. 2 |- ( ( ( x e. A |-> C ) |` A ) |` B ) = ( ( x e. A |-> C ) |` B )
6		inss1 3356	. . 3 |- ( A i^i B ) C_ A
7		resmpt 4956	. . 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 2206	1 |- ( ( x e. A |-> C ) |` B ) = ( x e. ( A i^i B ) |-> C )

Syntax hints:   = wceq 1353   i^i cin 3129   C_ wss 3130   |-> cmpt 4065   |` cres 4629

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175  ax-pr 4210

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2740  df-un 3134  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-opab 4066  df-mpt 4067  df-xp 4633  df-rel 4634  df-res 4639

This theorem is referenced by:  offres  6136