Theorem funexw 6080

Description: Weak version of funex 5708 that holds without ax-coll 4097. If the domain and codomain of a function exist, so does the function. (Contributed by Rohan Ridenour, 13-Aug-2023.)

Assertion

Ref	Expression
funexw	|- ( ( Fun F /\ dom F e. B /\ ran F e. C ) -> F e. _V )

Proof of Theorem funexw

Step	Hyp	Ref	Expression
1		xpexg 4718	. . 3 |- ( ( dom F e. B /\ ran F e. C ) -> ( dom F X. ran F ) e. _V )
2	1	3adant1 1005	. 2 |- ( ( Fun F /\ dom F e. B /\ ran F e. C ) -> ( dom F X. ran F ) e. _V )
3		funrel 5205	. . . 4 |- ( Fun F -> Rel F )
4		relssdmrn 5124	. . . 4 |- ( Rel F -> F C_ ( dom F X. ran F ) )
5	3, 4	syl 14	. . 3 |- ( Fun F -> F C_ ( dom F X. ran F ) )
6	5	3ad2ant1 1008	. 2 |- ( ( Fun F /\ dom F e. B /\ ran F e. C ) -> F C_ ( dom F X. ran F ) )
7	2, 6	ssexd 4122	1 |- ( ( Fun F /\ dom F e. B /\ ran F e. C ) -> F e. _V )

Syntax hints:   -> wi 4   /\ w3a 968   e. wcel 2136   _Vcvv 2726   C_ wss 3116   X. cxp 4602   dom cdm 4604   ran crn 4605   Rel wrel 4609   Fun wfun 5182

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-cnv 4612  df-dm 4614  df-rn 4615  df-fun 5190

This theorem is referenced by:  mptexw  6081  mpoexw  6181