Theorem fundmeng 6310

Description: A function is equinumerous to its domain. Exercise 4 of [Suppes] p. 98. (Contributed by NM, 17-Sep-2013.)

Assertion

Ref	Expression
fundmeng	|- ( ( F e. V /\ Fun F ) -> dom F ~~ F )

Proof of Theorem fundmeng

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		funeq 4941	. . . 4 |- ( x = F -> ( Fun x <-> Fun F ) )
2		dmeq 4553	. . . . 5 |- ( x = F -> dom x = dom F )
3		id 19	. . . . 5 |- ( x = F -> x = F )
4	2, 3	breq12d 3798	. . . 4 |- ( x = F -> ( dom x ~~ x <-> dom F ~~ F ) )
5	1, 4	imbi12d 232	. . 3 |- ( x = F -> ( ( Fun x -> dom x ~~ x ) <-> ( Fun F -> dom F ~~ F ) ) )
6		vex 2604	. . . 4 |- x e. _V
7	6	fundmen 6309	. . 3 |- ( Fun x -> dom x ~~ x )
8	5, 7	vtoclg 2658	. 2 |- ( F e. V -> ( Fun F -> dom F ~~ F ) )
9	8	imp 122	1 |- ( ( F e. V /\ Fun F ) -> dom F ~~ F )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1284   e. wcel 1433   class class class wbr 3785   dom cdm 4363   Fun wfun 4916   ~~ cen 6242

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-13 1444  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964  ax-un 4188

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-sbc 2816  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-uni 3602  df-int 3637  df-br 3786  df-opab 3840  df-mpt 3841  df-id 4048  df-xp 4369  df-rel 4370  df-cnv 4371  df-co 4372  df-dm 4373  df-rn 4374  df-iota 4887  df-fun 4924  df-fn 4925  df-f 4926  df-f1 4927  df-fo 4928  df-f1o 4929  df-fv 4930  df-en 6245

This theorem is referenced by:  fndmeng  6312  fundmfi  6389