Theorem mptexw 6092

Description: Weak version of mptex 5722 that holds without ax-coll 4104. 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 5236	. 2 |- Fun ( x e. A |-> B )
2		mptexw.1	. . 3 |- A e. _V
3		eqid 2170	. . . 4 |- ( x e. A |-> B ) = ( x e. A |-> B )
4	3	dmmptss 5107	. . 3 |- dom ( x e. A |-> B ) C_ A
5	2, 4	ssexi 4127	. 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 5657	. . . 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 4127	. 2 |- ran ( x e. A |-> B ) e. _V
11		funexw 6091	. 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 1332	1 |- ( x e. A |-> B ) e. _V

Syntax hints:   e. wcel 2141   A.wral 2448   _Vcvv 2730   C_ wss 3121   |-> cmpt 4050   dom cdm 4611   ran crn 4612   Fun wfun 5192

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-fv 5206

This theorem is referenced by: (None)