Theorem resfunexgALT 6761

Description: Alternate proof of resfunexg 6136, shorter but requiring ax-pow 4594 and ax-un 6588. (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 5138	. . . 4 |- ( B e. C -> dom ( A |` B ) e. _V )
2	1	adantl 467	. . 3 |- ( ( Fun A /\ B e. C ) -> dom ( A |` B ) e. _V )
3		df-ima 4858	. . . 4 |- ( A " B ) = ran ( A |` B )
4		funimaexg 5669	. . . 4 |- ( ( Fun A /\ B e. C ) -> ( A " B ) e. _V )
5	3, 4	syl5eqelr 2513	. . 3 |- ( ( Fun A /\ B e. C ) -> ran ( A |` B ) e. _V )
6	2, 5	jca 534	. 2 |- ( ( Fun A /\ B e. C ) -> ( dom ( A |` B ) e. _V /\ ran ( A |` B ) e. _V ) )
7		xpexg 6598	. 2 |- ( ( dom ( A |` B ) e. _V /\ ran ( A |` B ) e. _V ) -> ( dom ( A |` B ) X. ran ( A |` B ) ) e. _V )
8		relres 5143	. . . 4 |- Rel ( A |` B )
9		relssdmrn 5367	. . . 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 4562	. . 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 674	. 2 |- ( ( dom ( A |` B ) X. ran ( A |` B ) ) e. _V -> ( A |` B ) e. _V )
13	6, 7, 12	3syl 18	1 |- ( ( Fun A /\ B e. C ) -> ( A |` B ) e. _V )

Syntax hints:   -> wi 4   /\ wa 370   e. wcel 1867   _Vcvv 3078   C_ wss 3433   X. cxp 4843   dom cdm 4845   ran crn 4846   |` cres 4847   "cima 4848   Rel wrel 4850   Fun wfun 5586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-fun 5594

This theorem is referenced by: (None)