Theorem resfunexgALT 5881

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 5518 but requires ax-pow 4009 and ax-un 4260. (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 4736	. . . 4 |- ( B e. C -> dom ( A |` B ) e. _V )
2	1	adantl 271	. . 3 |- ( ( Fun A /\ B e. C ) -> dom ( A |` B ) e. _V )
3		df-ima 4451	. . . 4 |- ( A " B ) = ran ( A |` B )
4		funimaexg 5098	. . . 4 |- ( ( Fun A /\ B e. C ) -> ( A " B ) e. _V )
5	3, 4	syl5eqelr 2175	. . 3 |- ( ( Fun A /\ B e. C ) -> ran ( A |` B ) e. _V )
6	2, 5	jca 300	. 2 |- ( ( Fun A /\ B e. C ) -> ( dom ( A |` B ) e. _V /\ ran ( A |` B ) e. _V ) )
7		xpexg 4552	. 2 |- ( ( dom ( A |` B ) e. _V /\ ran ( A |` B ) e. _V ) -> ( dom ( A |` B ) X. ran ( A |` B ) ) e. _V )
8		relres 4741	. . . 4 |- Rel ( A |` B )
9		relssdmrn 4951	. . . 4 |- ( Rel ( A |` B ) -> ( A |` B ) C_ ( dom ( A |` B ) X. ran ( A |` B ) ) )
10	8, 9	ax-mp 7	. . 3 |- ( A |` B ) C_ ( dom ( A |` B ) X. ran ( A |` B ) )
11		ssexg 3978	. . 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 415	. 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 102   e. wcel 1438   _Vcvv 2619   C_ wss 2999   X. cxp 4436   dom cdm 4438   ran crn 4439   |` cres 4440   "cima 4441   Rel wrel 4443   Fun wfun 5009

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3954  ax-sep 3957  ax-pow 4009  ax-pr 4036  ax-un 4260

This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-uni 3654  df-br 3846  df-opab 3900  df-id 4120  df-xp 4444  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-rn 4449  df-res 4450  df-ima 4451  df-fun 5017

This theorem is referenced by: (None)