Theorem iprc 5046

Description: The identity function is a proper class. This means, for example, that we cannot use it as a member of the class of continuous functions unless it is restricted to a set. (Contributed by NM, 1-Jan-2007.)

Assertion

Ref	Expression
iprc	|- -. _I e. _V

Proof of Theorem iprc

Step	Hyp	Ref	Expression
1		vprc 4260	. . 3 |- -. _V e. _V
2		dmi 4991	. . . 4 |- dom _I = _V
3	2	eleq1i 2304	. . 3 |- ( dom _I e. _V <-> _V e. _V )
4	1, 3	mtbir 682	. 2 |- -. dom _I e. _V
5		dmexg 5041	. 2 |- ( _I e. _V -> dom _I e. _V )
6	4, 5	mto 672	1 |- -. _I e. _V

Syntax hints:   -. wn 3   e. wcel 2209   _Vcvv 2821   _I cid 4428   dom cdm 4769

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-13 2211  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-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-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-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-dm 4779  df-rn 4780

This theorem is referenced by: (None)