Theorem resfunexgALT 6008

Description: The restriction of a function to a set exists. Compare Proposition 6.17 of [TakeutiZaring] p. 28. This version has a shorter proof than resfunexg 5641 but requires ax-pow 4098 and ax-un 4355. (Contributed by NM, 7-Apr-1995.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
resfunexgALT	|- ( ( Fun A /\ B e. C ) -> ( A |` B ) e. _V )

Proof of Theorem resfunexgALT

Step	Hyp	Ref	Expression
1		dmresexg 4842	. . . 4 |- ( B e. C -> dom ( A |` B ) e. _V )
2	1	adantl 275	. . 3 |- ( ( Fun A /\ B e. C ) -> dom ( A |` B ) e. _V )
3		df-ima 4552	. . . 4 |- ( A " B ) = ran ( A |` B )
4		funimaexg 5207	. . . 4 |- ( ( Fun A /\ B e. C ) -> ( A " B ) e. _V )
5	3, 4	eqeltrrid 2227	. . 3 |- ( ( Fun A /\ B e. C ) -> ran ( A |` B ) e. _V )
6	2, 5	jca 304	. 2 |- ( ( Fun A /\ B e. C ) -> ( dom ( A |` B ) e. _V /\ ran ( A |` B ) e. _V ) )
7		xpexg 4653	. 2 |- ( ( dom ( A |` B ) e. _V /\ ran ( A |` B ) e. _V ) -> ( dom ( A |` B ) X. ran ( A |` B ) ) e. _V )
8		relres 4847	. . . 4 |- Rel ( A |` B )
9		relssdmrn 5059	. . . 4 |- ( Rel ( A |` B ) -> ( A |` B ) C_ ( dom ( A |` B ) X. ran ( A |` B ) ) )
10	8, 9	ax-mp 5	. . 3 |- ( A |` B ) C_ ( dom ( A |` B ) X. ran ( A |` B ) )
11		ssexg 4067	. . 3 |- ( ( ( A |` B ) C_ ( dom ( A |` B ) X. ran ( A |` B ) ) /\ ( dom ( A |` B ) X. ran ( A |` B ) ) e. _V ) -> ( A |` B ) e. _V )
12	10, 11	mpan 420	. 2 |- ( ( dom ( A |` B ) X. ran ( A |` B ) ) e. _V -> ( A |` B ) e. _V )
13	6, 7, 12	3syl 17	1 |- ( ( Fun A /\ B e. C ) -> ( A |` B ) e. _V )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 1480   _Vcvv 2686   C_ wss 3071   X. cxp 4537   dom cdm 4539   ran crn 4540   |` cres 4541   "cima 4542   Rel wrel 4544   Fun wfun 5117

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-fun 5125

This theorem is referenced by: (None)