Theorem fundmeng 7099

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 5400	. . . 4 |- ( x = F -> ( Fun x <-> Fun F ) )
2		dmeq 4983	. . . . 5 |- ( x = F -> dom x = dom F )
3		id 19	. . . . 5 |- ( x = F -> x = F )
4	2, 3	breq12d 4144	. . . 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 2824	. . . 4 |- x e. _V
7	6	fundmen 7098	. . 3 |- ( Fun x -> dom x ~~ x )
8	5, 7	vtoclg 2883	. 2 |- ( F e. V -> ( Fun F -> dom F ~~ F ) )
9	8	imp 124	1 |- ( ( F e. V /\ Fun F ) -> dom F ~~ F )

This proof depends on syntax axioms:   -> wi 4   /\ wa 104   = wceq 1402   e. wcel 2209   class class class wbr 4131   dom cdm 4776   Fun wfun 5374   ~~ cen 7024

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 4251  ax-pow 4313  ax-pr 4348  ax-un 4580

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-sbc 3052  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4132  df-opab 4194  df-mpt 4195  df-id 4440  df-xp 4782  df-rel 4783  df-cnv 4784  df-co 4785  df-dm 4786  df-rn 4787  df-iota 5340  df-fun 5382  df-fn 5383  df-f 5384  df-f1 5385  df-fo 5386  df-f1o 5387  df-fv 5388  df-en 7027

This theorem is used by:  fndmeng  7102  fundmfi  7255  usgrsizedgen  16659  upgr2wlkdc  16823