Theorem resfunexgALT 6269

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 5874 but requires ax-pow 4264 and ax-un 4530. (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 5036	. . . 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 4738	. . . 4 |- ( A " B ) = ran ( A |` B )
4		funimaexg 5414	. . . 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 4840	. 2 |- ( ( dom ( A |` B ) e. _V /\ ran ( A |` B ) e. _V ) -> ( dom ( A |` B ) X. ran ( A |` B ) ) e. _V )
8		relres 5041	. . . 4 |- Rel ( A |` B )
9		relssdmrn 5257	. . . 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 4228	. . 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 2802   C_ wss 3200   X. cxp 4723   dom cdm 4725   ran crn 4726   |` cres 4727   "cima 4728   Rel wrel 4730   Fun wfun 5320

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-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4204  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  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-uni 3894  df-br 4089  df-opab 4151  df-id 4390  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-fun 5328

This theorem is referenced by: (None)