Theorem fsetsspwxp 6948

Description: The class of all functions from A into B is a subclass of the power class of the cartesion product of A and B. (Contributed by AV, 13-Sep-2024.)

Assertion

Ref	Expression
fsetsspwxp	|- { f | f : A --> B } C_ ~P ( A X. B )

Distinct variable groups:   A, f   B, f

Proof of Theorem fsetsspwxp

Dummy variable g is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fssxp 5555	. . 3 |- ( g : A --> B -> g C_ ( A X. B ) )
2		vex 2824	. . . 4 |- g e. _V
3		feq1 5516	. . . 4 |- ( f = g -> ( f : A --> B <-> g : A --> B ) )
4	2, 3	elab 2970	. . 3 |- ( g e. { f | f : A --> B } <-> g : A --> B )
5		velpw 3695	. . 3 |- ( g e. ~P ( A X. B ) <-> g C_ ( A X. B ) )
6	1, 4, 5	3imtr4i 201	. 2 |- ( g e. { f | f : A --> B } -> g e. ~P ( A X. B ) )
7	6	ssriv 3252	1 |- { f | f : A --> B } C_ ~P ( A X. B )

This proof depends on syntax axioms:   e. wcel 2209   {cab 2224   C_ wss 3220   ~Pcpw 3688   X. cxp 4772   -->wf 5373

This proof depends on 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 4249  ax-pow 4311  ax-pr 4346

This proof 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 3690  df-sn 3715  df-pr 3716  df-op 3718  df-br 4131  df-opab 4193  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-fun 5379  df-fn 5380  df-f 5381

This theorem is used by: (None)