Theorem fundmeng 6797

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 5228	. . . 4 |- ( x = F -> ( Fun x <-> Fun F ) )
2		dmeq 4820	. . . . 5 |- ( x = F -> dom x = dom F )
3		id 19	. . . . 5 |- ( x = F -> x = F )
4	2, 3	breq12d 4011	. . . 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 2738	. . . 4 |- x e. _V
7	6	fundmen 6796	. . 3 |- ( Fun x -> dom x ~~ x )
8	5, 7	vtoclg 2795	. 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 1353   e. wcel 2146   class class class wbr 3998   dom cdm 4620   Fun wfun 5202   ~~ cen 6728

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-v 2737  df-sbc 2961  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-int 3841  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-rn 4631  df-iota 5170  df-fun 5210  df-fn 5211  df-f 5212  df-f1 5213  df-fo 5214  df-f1o 5215  df-fv 5216  df-en 6731

This theorem is referenced by:  fndmeng  6800  fundmfi  6927