Theorem resmpt3 5062

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 5025	. 2 |- ( ( ( x e. A |-> C ) |` A ) |` B ) = ( ( x e. A |-> C ) |` ( A i^i B ) )
2		ssid 3247	. . . 4 |- A C_ A
3		resmpt 5061	. . . 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 5009	. 2 |- ( ( ( x e. A |-> C ) |` A ) |` B ) = ( ( x e. A |-> C ) |` B )
6		inss1 3427	. . 3 |- ( A i^i B ) C_ A
7		resmpt 5061	. . 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 2260	1 |- ( ( x e. A |-> C ) |` B ) = ( x e. ( A i^i B ) |-> C )

Syntax hints:   = wceq 1397   i^i cin 3199   C_ wss 3200   |-> cmpt 4150   |` cres 4727

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-opab 4151  df-mpt 4152  df-xp 4731  df-rel 4732  df-res 4737

This theorem is referenced by:  mptima  5088  offres  6296