Theorem mapfset 6935

Description: If B is a set, the value of the set exponentiation ( B ^m A ) is the class of all functions from A to B. Generalisation of mapvalg 6922 to arbitrary domains. Note that the class { f | f : A --> B } can only contain set-functions, as opposed to arbitrary class-functions. When A is a proper class, there can be no set-functions on it, so the above class is empty (see also fsetdmprc0 6940), hence a set. In this case, both sides of the equality in this theorem are the empty set. (Contributed by AV, 8-Aug-2024.)

Assertion

Ref	Expression
mapfset	|- ( B e. V -> { f | f : A --> B } = ( B ^m A ) )

Distinct variable groups:   A, f   B, f

Allowed substitution hint:   V( f)

Proof of Theorem mapfset

Dummy variable m is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2824	. . . 4 |- m e. _V
2		feq1 5511	. . . 4 |- ( f = m -> ( f : A --> B <-> m : A --> B ) )
3	1, 2	elab 2970	. . 3 |- ( m e. { f | f : A --> B } <-> m : A --> B )
4		simpr 110	. . . . . . 7 |- ( ( m : A --> B /\ B e. V ) -> B e. V )
5		dmfex 5577	. . . . . . . . 9 |- ( ( m e. _V /\ m : A --> B ) -> A e. _V )
6	1, 5	mpan 428	. . . . . . . 8 |- ( m : A --> B -> A e. _V )
7	6	adantr 276	. . . . . . 7 |- ( ( m : A --> B /\ B e. V ) -> A e. _V )
8	4, 7	elmapd 6926	. . . . . 6 |- ( ( m : A --> B /\ B e. V ) -> ( m e. ( B ^m A ) <-> m : A --> B ) )
9	8	exbiri 382	. . . . 5 |- ( m : A --> B -> ( B e. V -> ( m : A --> B -> m e. ( B ^m A ) ) ) )
10	9	pm2.43b 52	. . . 4 |- ( B e. V -> ( m : A --> B -> m e. ( B ^m A ) ) )
11		elmapi 6934	. . . 4 |- ( m e. ( B ^m A ) -> m : A --> B )
12	10, 11	impbid1 142	. . 3 |- ( B e. V -> ( m : A --> B <-> m e. ( B ^m A ) ) )
13	3, 12	bitrid 192	. 2 |- ( B e. V -> ( m e. { f | f : A --> B } <-> m e. ( B ^m A ) ) )
14	13	eqrdv 2236	1 |- ( B e. V -> { f | f : A --> B } = ( B ^m A ) )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   {cab 2224   _Vcvv 2821   -->wf 5368  (class class class)co 6075   ^m cmap 6912

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-in1 623  ax-in2 624  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  ax-setind 4679

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  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-ne 2421  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-dif 3222  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-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-fv 5380  df-ov 6078  df-oprab 6079  df-mpo 6080  df-map 6914

This theorem is referenced by:  mapssfsetg  6936