Theorem resfunexgALT 6279

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 5883 but requires ax-pow 4270 and ax-un 4536. (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 5042	. . . 4 |- ( B e. C -> dom ( A |` B ) e. _V )
2	1	adantl 277	. . 3 |- ( ( Fun A /\ B e. C ) -> dom ( A |` B ) e. _V )
3		df-ima 4744	. . . 4 |- ( A " B ) = ran ( A |` B )
4		funimaexg 5421	. . . 4 |- ( ( Fun A /\ B e. C ) -> ( A " B ) e. _V )
5	3, 4	eqeltrrid 2319	. . 3 |- ( ( Fun A /\ B e. C ) -> ran ( A |` B ) e. _V )
6	2, 5	jca 306	. 2 |- ( ( Fun A /\ B e. C ) -> ( dom ( A |` B ) e. _V /\ ran ( A |` B ) e. _V ) )
7		xpexg 4846	. 2 |- ( ( dom ( A |` B ) e. _V /\ ran ( A |` B ) e. _V ) -> ( dom ( A |` B ) X. ran ( A |` B ) ) e. _V )
8		relres 5047	. . . 4 |- Rel ( A |` B )
9		relssdmrn 5264	. . . 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 4233	. . 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 424	. 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 104   e. wcel 2202   _Vcvv 2803   C_ wss 3201   X. cxp 4729   dom cdm 4731   ran crn 4732   |` cres 4733   "cima 4734   Rel wrel 4736   Fun wfun 5327

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-fun 5335

This theorem is referenced by: (None)