Theorem resmpt3 5027

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 4990	. 2 |- ( ( ( x e. A |-> C ) |` A ) |` B ) = ( ( x e. A |-> C ) |` ( A i^i B ) )
2		ssid 3221	. . . 4 |- A C_ A
3		resmpt 5026	. . . 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 4974	. 2 |- ( ( ( x e. A |-> C ) |` A ) |` B ) = ( ( x e. A |-> C ) |` B )
6		inss1 3401	. . 3 |- ( A i^i B ) C_ A
7		resmpt 5026	. . 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 2236	1 |- ( ( x e. A |-> C ) |` B ) = ( x e. ( A i^i B ) |-> C )

Syntax hints:   = wceq 1373   i^i cin 3173   C_ wss 3174   |-> cmpt 4121   |` cres 4695

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 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-opab 4122  df-mpt 4123  df-xp 4699  df-rel 4700  df-res 4705

This theorem is referenced by:  mptima  5053  offres  6243