Theorem fundmeng 6863

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 5275	. . . 4 |- ( x = F -> ( Fun x <-> Fun F ) )
2		dmeq 4863	. . . . 5 |- ( x = F -> dom x = dom F )
3		id 19	. . . . 5 |- ( x = F -> x = F )
4	2, 3	breq12d 4043	. . . 4 |- ( x = F -> ( dom x ~~ x <-> dom F ~~ F ) )
5	1, 4	imbi12d 234	. . 3 |- ( x = F -> ( ( Fun x -> dom x ~~ x ) <-> ( Fun F -> dom F ~~ F ) ) )
6		vex 2763	. . . 4 |- x e. _V
7	6	fundmen 6862	. . 3 |- ( Fun x -> dom x ~~ x )
8	5, 7	vtoclg 2821	. 2 |- ( F e. V -> ( Fun F -> dom F ~~ F ) )
9	8	imp 124	1 |- ( ( F e. V /\ Fun F ) -> dom F ~~ F )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1364   e. wcel 2164   class class class wbr 4030   dom cdm 4660   Fun wfun 5249   ~~ cen 6794

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-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2987  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-br 4031  df-opab 4092  df-mpt 4093  df-id 4325  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-en 6797

This theorem is referenced by:  fndmeng  6866  fundmfi  6998