Theorem fundmeng 6899

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 5291	. . . 4 |- ( x = F -> ( Fun x <-> Fun F ) )
2		dmeq 4878	. . . . 5 |- ( x = F -> dom x = dom F )
3		id 19	. . . . 5 |- ( x = F -> x = F )
4	2, 3	breq12d 4057	. . . 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 2775	. . . 4 |- x e. _V
7	6	fundmen 6898	. . 3 |- ( Fun x -> dom x ~~ x )
8	5, 7	vtoclg 2833	. 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 1373   e. wcel 2176   class class class wbr 4044   dom cdm 4675   Fun wfun 5265   ~~ cen 6825

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-pow 4218  ax-pr 4253  ax-un 4480

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4045  df-opab 4106  df-mpt 4107  df-id 4340  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-en 6828

This theorem is referenced by:  fndmeng  6902  fundmfi  7039