Theorem mptexw 6315

Description: Weak version of mptex 5917 that holds without ax-coll 4230. If the domain and codomain of a function given by maps-to notation are sets, the function is a set. (Contributed by Rohan Ridenour, 13-Aug-2023.)

Hypotheses

Ref	Expression
mptexw.1	|- A e. _V
mptexw.2	|- C e. _V
mptexw.3	|- A. x e. A B e. C

Assertion

Ref	Expression
mptexw	|- ( x e. A |-> B ) e. _V

Distinct variable groups:   x, A   x, C

Allowed substitution hint:   B( x)

Proof of Theorem mptexw

Step	Hyp	Ref	Expression
1		funmpt 5395	. 2 |- Fun ( x e. A |-> B )
2		mptexw.1	. . 3 |- A e. _V
3		eqid 2234	. . . 4 |- ( x e. A |-> B ) = ( x e. A |-> B )
4	3	dmmptss 5264	. . 3 |- dom ( x e. A |-> B ) C_ A
5	2, 4	ssexi 4253	. 2 |- dom ( x e. A |-> B ) e. _V
6		mptexw.2	. . 3 |- C e. _V
7		mptexw.3	. . . 4 |- A. x e. A B e. C
8	3	rnmptss 5843	. . . 4 |- ( A. x e. A B e. C -> ran ( x e. A |-> B ) C_ C )
9	7, 8	ax-mp 5	. . 3 |- ran ( x e. A |-> B ) C_ C
10	6, 9	ssexi 4253	. 2 |- ran ( x e. A |-> B ) e. _V
11		funexw 6314	. 2 |- ( ( Fun ( x e. A |-> B ) /\ dom ( x e. A |-> B ) e. _V /\ ran ( x e. A |-> B ) e. _V ) -> ( x e. A |-> B ) e. _V )
12	1, 5, 10, 11	mp3an 1374	1 |- ( x e. A |-> B ) e. _V

Syntax hints:   e. wcel 2205   A.wral 2522   _Vcvv 2815   C_ wss 3214   |-> cmpt 4176   dom cdm 4754   ran crn 4755   Fun wfun 5351

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 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-sbc 3046  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fv 5365

This theorem is referenced by: (None)