Theorem funexw 6331

Description: Weak version of funex 5931 that holds without ax-coll 4241. 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 4884	. . 3 |- ( ( dom F e. B /\ ran F e. C ) -> ( dom F X. ran F ) e. _V )
2	1	3adant1 1046	. 2 |- ( ( Fun F /\ dom F e. B /\ ran F e. C ) -> ( dom F X. ran F ) e. _V )
3		funrel 5389	. . . 4 |- ( Fun F -> Rel F )
4		relssdmrn 5303	. . . 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 1049	. 2 |- ( ( Fun F /\ dom F e. B /\ ran F e. C ) -> F C_ ( dom F X. ran F ) )
7	2, 6	ssexd 4268	1 |- ( ( Fun F /\ dom F e. B /\ ran F e. C ) -> F e. _V )

Syntax hints:   -> wi 4   /\ w3a 1009   e. wcel 2209   _Vcvv 2821   C_ wss 3220   X. cxp 4767   dom cdm 4769   ran crn 4770   Rel wrel 4774   Fun wfun 5366

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-opab 4188  df-xp 4775  df-rel 4776  df-cnv 4777  df-dm 4779  df-rn 4780  df-fun 5374

This theorem is referenced by:  mptexw  6332  mpoexw  6439